1.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone

0 (number)
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Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus

0 (number)

0 (number)
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The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.

0 (number)
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The back of Olmec stela C from Tres Zapotes, the second oldest Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.

2.
Factorization
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In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 ×5, in all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viètes formulas relate the coefficients of a polynomial to its roots, the opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms. Integer factorization for large integers appears to be a difficult problem, there is no known method to carry it out quickly. Its complexity is the basis of the security of some public key cryptography algorithms. A matrix can also be factorized into a product of matrices of special types, One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types, QR decomposition, LQ, QL, RQ and this situation is generalized by factorization systems. By the fundamental theorem of arithmetic, every integer greater than 1 has a unique prime factorization. Given an algorithm for integer factorization, one can factor any integer down to its constituent primes by repeated application of this algorithm, for very large numbers, no efficient classical algorithm is known. Modern techniques for factoring polynomials are fast and efficient, but use sophisticated mathematical ideas and these techniques are used in the construction of computer routines for carrying out polynomial factorization in Computer algebra systems. This article is concerned with classical techniques. While the general notion of factoring just means writing an expression as a product of simpler expressions, when factoring polynomials this means that the factors are to be polynomials of smaller degree. Thus, while x 2 − y = is a factorization of the expression, another issue concerns the coefficients of the factors. It is not always possible to do this, and a polynomial that can not be factored in this way is said to be irreducible over this type of coefficient, thus, x2 -2 is irreducible over the integers and x2 +4 is irreducible over the reals. In the first example, the integers 1 and -2 can also be thought of as real numbers, and if they are, then x 2 −2 = shows that this polynomial factors over the reals. Similarly, since the integers 1 and 4 can be thought of as real and hence complex numbers, x2 +4 splits over the complex numbers, i. e. x 2 +4 =. The fundamental theorem of algebra can be stated as, Every polynomial of n with complex number coefficients splits completely into n linear factors

Factorization
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A visual representation of the factorization of cubes using volumes. For a sum of cubes, simply substitute z=-y.

3.
Ternary numeral system
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The ternary numeral system has three as its base. Analogous to a bit, a digit is a trit. One trit is equivalent to bits of information. Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary, for example, decimal 365 corresponds to binary 101101101 and to ternary 111112. However, they are far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary. The value of a number with n bits that are all 1 is 2n −1. Then N = M, N = /, and N = bd −1, for a three-digit ternary number, N =33 −1 =26 =2 ×32 +2 ×31 +2 ×30 =18 +6 +2. Nonary or septemvigesimal can be used for representation of ternary. A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a hand for counting prayers. In certain analog logic, the state of the circuit is often expressed ternary and this is most commonly seen in Transistor–transistor logic using 7406 open collector logic. The output is said to either be low, high, or open, in this configuration the output of the circuit is actually not connected to any voltage reference at all. Where the signal is usually grounded to a reference, or at a certain voltage level. Thus, the voltage level is sometimes unpredictable. A rare ternary point is used to denote fractional parts of an inning in baseball, since each inning consists of three outs, each out is considered one third of an inning and is denoted as.1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning 3⅔. In this usage, only the part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like the Sierpinski triangle or the Cantor set conveniently, additionally, it turns out that the ternary representation is useful for defining the Cantor set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 that have an expression that does not contain any instance of the digit 1

Ternary numeral system
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Numeral systems

4.
Quaternary numeral system
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Quaternary is the base-4 numeral system. It uses the digits 0,1,2 and 3 to represent any real number. Four is the largest number within the range and one of two numbers that is both a square and a highly composite number, making quaternary a convenient choice for a base at this scale. Despite being twice as large, its economy is equal to that of binary. However, it no better in the localization of prime numbers. See decimal and binary for a discussion of these properties, as with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4,8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2,3 or 4 binary digits, for example, in base 4,302104 =11001001002. Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, by analogy with byte and nybble, a quaternary digit is sometimes called a crumb. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca, the Kharosthi numerals have a partial base 4 counting system from 1 to decimal 10. Quaternary numbers are used in the representation of 2D Hilbert curves, here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected, parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0,1,2. With this encoding, the complementary digit pairs 0↔3, and 1↔2 match the complementation of the pairs, A↔T and C↔G. For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010, quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits

Quaternary numeral system
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Numeral systems

5.
Duodecimal
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The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated 2 and this notation was introduced by Sir Isaac Pitman. These digit forms are available as Unicode characters on computerized systems since June 2015 as ↊ and ↋, other notations use A, T, or X for ten and B or E for eleven. The number twelve is written as 10 in duodecimal, whereas the digit string 12 means 1 dozen and 2 units. Similarly, in duodecimal 100 means 1 gross,1000 means 1 great gross, the number twelve, a superior highly composite number, is the smallest number with four non-trivial factors, and the smallest to include as factors all four numbers within the subitizing range. As a result, duodecimal has been described as the number system. Of its factors,2 and 3 are prime, which means the reciprocals of all 3-smooth numbers have a representation in duodecimal. In particular, the five most elementary fractions all have a terminating representation in duodecimal. This all makes it a convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary. Although the trigesimal and sexagesimal systems do even better in respect, this is at the cost of unwieldy multiplication tables. In this section, numerals are based on decimal places, for example,10 means ten,12 means twelve. Languages using duodecimal number systems are uncommon, germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif, historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in a foot,12 troy ounces in a troy pound,12 old British pence in a shilling,24 hours in a day. The Romans used a system based on 12, including the uncia which became both the English words ounce and inch. The importance of 12 has been attributed to the number of cycles in a year. It is possible to count to 12 with the acting as a pointer

Duodecimal
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Numeral systems

Duodecimal
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A duodecimal multiplication table

6.
Hexadecimal
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In mathematics and computing, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, Hexadecimal numerals are widely used by computer system designers and programmers. As each hexadecimal digit represents four binary digits, it allows a more human-friendly representation of binary-coded values, one hexadecimal digit represents a nibble, which is half of an octet or byte. For example, a byte can have values ranging from 00000000 to 11111111 in binary form. In a non-programming context, a subscript is typically used to give the radix, several notations are used to support hexadecimal representation of constants in programming languages, usually involving a prefix or suffix. The prefix 0x is used in C and related languages, where this value might be denoted as 0x2AF3, in contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously, a numerical subscript can give the base explicitly,15910 is decimal 159,15916 is hexadecimal 159, which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex, or 159d and 159h. example. com/name%20with%20spaces where %20 is the space character, thus ’, represents the right single quotation mark, Unicode code point number 2019 in hex,8217. In the Unicode standard, a value is represented with U+ followed by the hex value. Color references in HTML, CSS and X Window can be expressed with six hexadecimal digits prefixed with #, white, CSS allows 3-hexdigit abbreviations with one hexdigit per component, #FA3 abbreviates #FFAA33. *nix shells, AT&T assembly language and likewise the C programming language, to output an integer as hexadecimal with the printf function family, the format conversion code %X or %x is used. In Intel-derived assembly languages and Modula-2, hexadecimal is denoted with a suffixed H or h, some assembly languages use the notation HABCD. Ada and VHDL enclose hexadecimal numerals in based numeric quotes, 16#5A3#, for bit vector constants VHDL uses the notation x5A3. Verilog represents hexadecimal constants in the form 8hFF, where 8 is the number of bits in the value, the Smalltalk language uses the prefix 16r, 16r5A3 PostScript and the Bourne shell and its derivatives denote hex with prefix 16#, 16#5A3. For PostScript, binary data can be expressed as unprefixed consecutive hexadecimal pairs, in early systems when a Macintosh crashed, one or two lines of hexadecimal code would be displayed under the Sad Mac to tell the user what went wrong. Common Lisp uses the prefixes #x and #16r, setting the variables *read-base* and *print-base* to 16 can also used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, MSX BASIC, QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with &H, &H5A3 BBC BASIC and Locomotive BASIC use & for hex. TI-89 and 92 series uses a 0h prefix, 0h5A3 ALGOL68 uses the prefix 16r to denote hexadecimal numbers, binary, quaternary and octal numbers can be specified similarly

Hexadecimal
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Numeral systems

Hexadecimal
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Bruce Alan Martin's hexadecimal notation proposal

Hexadecimal
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Hexadecimal finger-counting scheme.

7.
Vigesimal
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The vigesimal or base 20 numeral system is based on twenty. In a vigesimal system, twenty individual numerals are used. One modern method of finding the extra needed symbols is to write ten as the letter A20, to write nineteen as J20, and this is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters A–F. Another method skips over the letter I, in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, the number twenty is written as 1020. According to this notation,2020 means forty in decimal = + D020 means two hundred and sixty in decimal = +10020 means four hundred in decimal = + +, in the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example,10 means ten,20 means twenty, in decimal, dividing by three twice only gives one digit periods because 9 is the number below ten. 21, however, the adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods, the prime factorization of twenty is 22 ×5, so it is not a perfect power. However, its part,5, is congruent to 1. Thus, according to Artins conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37. 395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a set of bases found that, of the first 15,456 primes. In many European languages,20 is used as a base, vigesimal systems are common in Africa, for example in Yoruba. Ogún,20, is the basic numeric block, ogójì,40, =20 multiplied by 2. Ogota,60, =20 multiplied by 3, ogorin,80, =20 multiplied by 4. Ogorun,100, =20 multiplied by 5, twenty was a base in the Maya and Aztec number systems. The Maya used the names for the powers of twenty, kal, bak, pic, calab, kinchil. See also Maya numerals and Maya calendar, Mayan languages, Yucatec, the Aztec called them, cempoalli, centzontli, cenxiquipilli, cempoalxiquipilli, centzonxiquipilli and cempoaltzonxiquipilli. Note that the ce prefix at the beginning means one and is replaced with the number to get the names of other multiples of the power

Vigesimal
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Numeral systems

Vigesimal
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The Maya numerals are a base-20 system.

8.
Euro
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Outside of Europe, a number of overseas territories of EU members also use the euro as their currency. Additionally,210 million people worldwide as of 2013 use currencies pegged to the euro, the euro is the second largest reserve currency as well as the second most traded currency in the world after the United States dollar. The name euro was adopted on 16 December 1995 in Madrid. The euro was introduced to world markets as an accounting currency on 1 January 1999. While the euro dropped subsequently to US$0.8252 within two years, it has traded above the U. S. dollar since the end of 2002, peaking at US$1.6038 on 18 July 2008. In July 2012, the euro fell below US$1.21 for the first time in two years, following concerns raised over Greek debt and Spains troubled banking sector, as of 26 March 2017, the euro–dollar exchange rate stands at ~ US$1.07. The euro is managed and administered by the Frankfurt-based European Central Bank, as an independent central bank, the ECB has sole authority to set monetary policy. The Eurosystem participates in the printing, minting and distribution of notes and coins in all states. The 1992 Maastricht Treaty obliges most EU member states to adopt the euro upon meeting certain monetary and budgetary convergence criteria, all nations that have joined the EU since 1993 have pledged to adopt the euro in due course. Since 5 January 2002, the central banks and the ECB have issued euro banknotes on a joint basis. Euro banknotes do not show which central bank issued them, Eurosystem NCBs are required to accept euro banknotes put into circulation by other Eurosystem members and these banknotes are not repatriated. The ECB issues 8% of the value of banknotes issued by the Eurosystem. In practice, the ECBs banknotes are put into circulation by the NCBs and these liabilities carry interest at the main refinancing rate of the ECB. The euro is divided into 100 cents, in Community legislative acts the plural forms of euro and cent are spelled without the s, notwithstanding normal English usage. Otherwise, normal English plurals are used, with many local variations such as centime in France. All circulating coins have a side showing the denomination or value. Due to the plurality in the European Union, the Latin alphabet version of euro is used. For the denominations except the 1-, 2- and 5-cent coins, beginning in 2007 or 2008 the old map is being replaced by a map of Europe also showing countries outside the Union like Norway

Euro
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The central bank has its seat in Frankfurt (Germany) and is in charge of the monetary policy of the euro area.

Euro
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Euro banknotes as of 2015

Euro
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Euro coins and banknotes of various denominations.

9.
Sprint Cup Series
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The Monster Energy NASCAR Cup Series is the top racing series of the National Association for Stock Car Auto Racing. It is named for the current sponsor, Monster Energy, but has been known by names in the past. The series began in 1949 as the Strictly Stock Series, in 1971, when the series began leasing its naming rights to the R. J. Reynolds Tobacco Company, it was referred to as the Winston Cup Series. A similar deal was made with Nextel in 2003, and it became the Nextel Cup Series, Sprint acquired Nextel in 2005, and in 2008 the series was renamed the Sprint Cup Series, which lasted until 2016. In December 2016, it was announced that Monster Energy would become the new sponsor starting in 2017. The championship is determined by a system, with points being awarded according to finish placement. The season is divided into two segments, after the first 26 races,16 drivers, selected primarily on the basis of wins during the first 26 races, are seeded based on their total number of wins. They compete in the last ten races, where the difference in points is greatly minimized and this is called the NASCAR playoffs. The series holds strong roots in the Southeastern United States, with half of the races in the 36-race season being held in that region, the current schedule includes tracks from around the United States. Regular season races were held in Canada, and exhibition races were held in Japan. The Daytona 500, the most prestigious race, had an audience of about 16 million U. S. viewers in 2009. Cup Series cars are unique in automobile racing, the engines are powerful enough to reach speeds of over 200 mph, but their weight coupled with a relatively simple aerodynamic package make for poor handling. The bodies and chassis of the cars are strictly regulated to ensure parity, in 1949, NASCAR introduced the Strictly Stock division, after sanctioning Modified and Roadster division races in 1948. Eight races were run on seven dirt ovals and on the Daytona Beach beach/street course, the first NASCAR Strictly Stock race was held at Charlotte Speedway on June 19,1949. Jim Roper was declared the winner of race after Glenn Dunaway was disqualified for having altered the rear springs on his car. The division was renamed Grand National for the 1950 season, reflecting NASCARs intent to make the more professional. It retained this name until 1971, the 1949 Strictly Stock season is regarded in NASCARs record books as the first season of GN/Cup history. Martinsville Speedway is the track on the 1949 schedule that remains on the current schedule

Sprint Cup Series
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Seven-time Winston Cup champion Richard Petty.

Sprint Cup Series
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Sprint Cup Series

Sprint Cup Series
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Seven-time Winston Cup champion Dale Earnhardt

Sprint Cup Series
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Six-time Sprint Cup Series champion, Jimmie Johnson.

10.
15 (number)
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15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name, in spoken English, the numbers 15 and 50 are often confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed,15 /fɪfˈtiːn/ vs 50 /ˈfɪfti/, however, in dates such as 1500 or when contrasting numbers in the teens, the stress generally shifts to the first syllable,15 /ˈfɪftiːn/. In a 24-hour clock, the hour is in conventional language called three or three oclock. A composite number, its divisors being 1,3 and 5. A repdigit in binary and quaternary, in hexadecimal, as well as all higher bases,15 is represented as F. the 4th discrete semiprime and the first member of the discrete semiprime family. It is thus the first odd discrete semiprime, the number proceeding 15,14 is itself a discrete semiprime and this is the first such pair of discrete semiprimes. The next example is the pair commencing 21, the smallest number that can be factorized using Shors quantum algorithm. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet, the aliquot sum of 15 is 9, a square prime 15 has an aliquot sequence of 6 members. 15 is the composite number in the 3-aliquot tree. The abundant 12 is also a member of this tree, fifteen is the aliquot sum of the consecutive 4-power 16, and the discrete semiprime 33. 15 and 16 form a Ruth-Aaron pair under the definition in which repeated prime factors are counted as often as they occur. There are 15 solutions to Známs problem of length 7, if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems. Group 15 of the table are sometimes known as the pnictogens. 15 Madadgar is designated as a number in Pakistan, for mobile phones, similar to the international GSM emergency number 112, if 112 is used in Pakistan. 112 can be used in an emergency if the phone is locked. The Hanbali Sunni madhab states that the age of fifteen of a solar or lunar calendar is when ones taklif begins and is the stage whereby one has his deeds recorded. In the Hebrew numbering system, the number 15 is not written according to the method, with the letters that represent 10 and 5

15 (number)
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Fifteen total individuals in the mollusk species Donax variabilis comprise the entire coloration and patterning in their phenotypes.

15 (number)
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The 15 perfect matchings of K 6

11.
19 (number)
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19 is the natural number following 18 and preceding 20. In a 24-hour clock, the hour is in conventional language called seven or seven oclock. 19 is the 8th prime number, the sequence continues 23,29,31,37. 19 is the seventh Mersenne prime exponent,19 is the fifth happy number and the third happy prime. 19 is the sum of two odd discrete semiprimes,65 and 77 and is the base of the 19-aliquot tree. 19 is the number of fourth powers needed to sum up to any natural number. It is the value of g.19 is the lowest prime centered triangular number, a centered hexagonal number. The only non-trivial normal magic hexagon contains 19 hexagons,19 is the first number with more than one digit that can be written from base 2 to base 19 using only the digits 0 to 9, the other number is 20. 19 is The TCP/IP port used for chargen, astronomy, Every 19 years, the solar year and the lunar year align in whats known as the metonic cycle. Quran code, There have been claims that patterns of the number 19 are present a number of times in the Quran. The Number of Verse and Sura together in the Quran which announces Jesus son of Maryams birth, in the Bábí and Baháí faiths, a group of 19 is called a Váhid, a Unity. The numerical value of this word in the Abjad numeral system is 19, the Baháí calendar is structured such that a year contains 19 months of 19 days each, as well as a 19-year cycle and a 361-year supercycle. The Báb and his disciples formed a group of 19, There were 19 Apostles of Baháulláh. With a similar name and anti-Vietnam War theme, I Was Only Nineteen by the Australian group Redgum reached number one on the Australian charts in 1983, in 2005 a hip hop version of the song was produced by The Herd. 19 is the name of Adeles 2008 debut album, so named since she was 19 years old at the time, hey Nineteen is a song by American jazz rock band Steely Dan, written by members Walter Becker and Donald Fagen, and released on their 1980 album Gaucho. Nineteen has been used as an alternative to twelve for a division of the octave into equal parts and this idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1/3 comma meantone. Some organs use the 19th harmonic to approximate a minor third and they refer to the ka-tet of 19, Directive Nineteen, many names add up to 19,19 seems to permeate every aspect of Roland and his travelers lives. In addition, the ends up being a powerful key

19 (number)
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A 19x19 Go board

19 (number)
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19 is a centered triangular number

12.
31 (number)
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31 is the natural number following 30 and preceding 32. As a Mersenne prime,31 is related to the perfect number 496,31 is also the 4th lucky prime and the 11th supersingular prime. 31 is a triangular number, the lowest prime centered pentagonal number. For the Steiner tree problem,31 is the number of possible Steiner topologies for Steiner trees with 4 terminals, at 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31,31 is a repdigit in base 5, and base 2. The numbers 31,331,3331,33331,333331,3333331, for a time it was thought that every number of the form 3w1 would be prime. Here,31 divides every fifteenth number in 3w1, the atomic number of gallium Messier object M31, a magnitude 4.5 galaxy in the constellation Andromeda. It is also known as the Andromeda Galaxy, and is visible to the naked eye in a modestly dark sky. The New General Catalogue object NGC31, a galaxy in the constellation Phoenix The Saros number of the solar eclipse series which began on -1805 January 31. The duration of Saros series 31 was 1316.2 years, the Saros number of the lunar eclipse series which began on -1774 May 30 and ended on -476 July 17. The duration of Saros series 31 was 1298.1 years, the jersey number 31 has been retired by several North American sports teams in honor of past playing greats, In Major League Baseball, The San Diego Padres, for Dave Winfield. The Chicago Cubs, for Ferguson Jenkins and Greg Maddux, the Atlanta Braves, also for Maddux. The New York Mets, for Mike Piazza, in the NBA, The Boston Celtics, for Cedric Maxwell. The Indiana Pacers, for Reggie Miller, in the NHL, The Edmonton Oilers, for Grant Fuhr. The New York Islanders, for Billy Smith, in the NFL, The Atlanta Falcons, for William Andrews. The New Orleans Saints, for Jim Taylor, NASCAR driver Jeff Burton drives #31, a car which was subject to a controversy when one of the sponsors changed its name after merging with another company. In ice hockey goaltenders often wear the number 31, in football the number 31 has been retired by Queens Park Rangers F. C.31 from the Prime Pages

31 (number)
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31 is a centered pentagonal number

13.
37 (number)
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37 is the natural number following 36 and preceding 38. Thirty-seven is the 12th prime number, a prime with 73. It is a hexagonal number and a star number. Every positive integer is the sum of at most 37 fifth powers,37 appears in the Padovan sequence, preceded by the terms 16,21, and 28. Since the greatest prime factor of 372 +1 =1370 is 137, the atomic number of rubidium The normal human body temperature in degrees Celsius Messier object M37, a magnitude 6. The duration of Saros series 37 was 1298.1 years, the Saros number of the lunar eclipse series which began on -1492 April 3 and ended on -194 May 22. The duration of Saros series 37 was 1298.1 years, kepler-37b is the smallest known planet. The New York Yankees, also for Stengel and this honor made him the first manager to have had his number retired by two different teams. In the NFL, The Detroit Lions, for Doak Walker, the San Francisco 49ers, for Jimmy Johnson. Thirty-seven is, The number of plays William Shakespeare is thought to have written, today the +37 prefix is shared by Lithuania, Latvia, Estonia, Moldova, Armenia, Belarus, Andorra, Monaco, San Marino and Vatican City. A television channel reserved for radio astronomy in the United States The number people are most likely to state when asked to give a number between 0 and 100. The inspiration for the album 37 Everywhere by Punchline List of highways numbered 37 Number Thirty-Seven, Pennsylvania, unincorporated community in Cambria County, Pennsylvania I37

37 (number)
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House number in Baarle (in its Belgian part)

14.
38 (number)
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38 is the natural number following 37 and preceding 39. 38 is the 11th distinct semiprime and the 7th in the family and it is the initial member of the third distinct semiprime pair. 38 has a sum of 22 which is itself a distinct semiprime In fact 38 is the first number to be at the head of a chain of four distinct semiprimes in its 8-member aliquot sequence. 38 is the 8th member of the 7-aliquot tree, −1 yields 523022617466601111760007224100074291199999999, which is the 16th factorial prime. There is no answer to the equation φ =38, making 38 a nontotient,38 is the sum of the squares of the first three primes. 37 and 38 are the first pair of positive integers not divisible by any of their digits. 38 is the largest even number which cannot be written as the sum of two odd composite numbers, there are only two normal magic hexagons, order 1 and order 3. The sum of row of an order 3 magic hexagon is 38. The duration of Saros series 38 was 1298.1 years, the lunar eclipse series which began on -1408 April 16 and ended on -111 June 3. The duration of Saros series 38 was 1298.1 years, the New General Catalogue object NGC38, a spiral galaxy in the constellation Pisces Thirty-eight is also, The 38th parallel north is the pre-Korean War boundary between North Korea and South Korea

38 (number)
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Most people will see the number 38, but people with red-green color blindness might see 88 instead.

15.
45 (number)
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45 is the natural number following 44 and followed by 46. Forty-five is a number, and in particular the sum of all the decimal digits. It is the smallest triangle number which can be written as the sum of two squares and it is also a hexagonal and 16-gonal number. 45 is the positive integer with a prime factorization of the form p2q. 33 is the sum of 45 and the aliquot sequence of 45 is. Since the greatest prime factor of 452 +1 =2026 is 1013, in base 10, it is a Kaprekar number and a Harshad number. The duration of Saros series 45 was 1280.1 years, the Saros number of the lunar eclipse series which began on −1369 August 19 and ended on 182 March. The duration of Saros series 45 was 1550.6 years, a card game, Forty-five.45, a 2006 motion picture. In the United States,45 is often a reference to one of two specific.45 caliber cartridges— the.45 Colt or the.45 ACP, in years of marriage, the sapphire wedding anniversary. Forty Five a Big Finish 2008 audio play made for the forty fifth anniversary of the British science fiction television show Doctor Who, issue 45 of The North Briton was thought to be seditious but its publisher, John Wilkes, was celebrated as a champion of liberty. The number 45 was used as a symbol of support for him, banquets were held with a theme of 45 while many items were produced showing the number or featuring it in some way. For example, a wig was produced with 45 curls

45 (number)
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45 rpm gramophone record

16.
64 (number)
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64 is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube of 4, and it is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime,64 is the sum of Eulers totient function for the first fourteen integers. It is also a number and a centered triangular number. 64 is also the first whole number that is both a square and a perfect cube. Since it is possible to find sequences of 64 consecutive integers such that each member shares a factor with either the first or the last member,64 is an Erdős–Woods number. In base 10, no integer added up to its own digits yields 64,64 is a superperfect number—a number such that σ = 2n. The New General Catalogue object NGC64, a spiral galaxy in the constellation Cetus. The Saros number of the solar eclipse series began on -832 April 11. The duration of Saros series 64 was 1262.1 years, the Saros number of the lunar eclipse series which began on -801 August 9 and ended on 714 February. The duration of Saros series 64 was 1514.5 years, route 64, a highway from Arizona to North Carolina The subject of the Beatles song When Im Sixty-Four The registry of the U. S. They include, singing, dancing, painting, poetry, playing cards, making arguments, making flower garlands, the number of the French department Pyrénées-Atlantiques The number of crayons in the popular Crayola 64 pack Base 64 is used for example with Base64 encoding. Maximum stack size in the video game Minecraft, in non-command situations,64 is a character in the Donald Duck comics universe. Number of codons in the RNA codon table under genetic code, number of golden disks in the myth of the Tower of Hanoi. A number referring to Tiananmen Square protests of 1989 in Chinese 64 is the title of a song by the hip-hop group Mellowhype from their album Blackenedwhite The 64,000 Dollar Question

64 (number)
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A chessboard has 64 squares

17.
66 (number)
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66 is the natural number following 65 and preceding 67. Usages of this include,66 is, a sphenic number. A semiperfect number, being a multiple of a perfect number, an Erdős–Woods number, since it is possible to find sequences of 66 consecutive integers such that each inner member shares a factor with either the first or the last member. Palindromic and a repdigit in bases 10,21 and 32 a Harshad number in bases 2,4,5,6,7,8,11,12,13 and 16. Messier object Spiral Galaxy M66, a magnitude 10.0 galaxy in the constellation Leo, the New General Catalogue object NGC66, a peculiar barred spiral galaxy in the constellation Cetus. The Saros number of the solar eclipse series began on 12 March -756. The duration of Saros series 66 was 1298.1 years, the Saros number of the lunar eclipse series which began on 12 August -671 and ended on 27 January 826. The duration of Saros series 66 was 1496.5 years, the atomic number of dysprosium, a lanthanide. 66 megahertz is a common divisor for the front side bus speed, overall central processing unit speed, and base bus speed. On a Core 2 CPU, and a Core 2 motherboard, the FSB is 1066 MHz, the speed is usually 666.67 MHz. The designation of the historic U. S. Route 66, the designation of US Interstate 66, a freeway that runs from Virginia to Washington, D. C. The designation of 36 US state and 2 territorial highways, phillips 66, a brand of gasoline and service station in the United States. The total number of chapters in the Bible book of Isaiah, the number of verses in Chapter 3 of the book of Lamentations in the Old Testament. The total number of books in the Protestant edition of the Bible combined, in Abjad numerals, The Name Of Allah numeric value is 66. The Green Bay Packers of the National Football League retired jersey number 66 for linebacker Ray Nitschke, the last Packer to wear number 66 before it was retired in 1983 to honor Nitschke was offensive tackle Larry Pfohl, who is better known to professional wrestling fans as Lex Luger. Jersey number 66 was retired by the Pittsburgh Penguins of the National Hockey League in honor of Mario Lemieux, sixty Six is a 2006 British movie about a bar mitzvah in London on the day of the 1966 World Cup final. In the Star Wars movie series, Order 66 is an order to the clone troopers to kill the Jedi commanding them. Route 66 was a popular US television series on CBS from 1960 to 1964, in the video game Fullmetal Alchemist, elusive villain Barry the Chopper is imprisoned in cell number 66, which later becomes his alias when battling the brothers at Laboratory Five

66 (number)
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Route 66 sign

18.
76 (number)
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76 is the natural number following 75 and preceding 77. A palindromic number and a repdigit in bases 18 and 37, a Harshad number in bases 4 and 5. An Erdős–Woods number since it is possible to find sequences of 76 consecutive integers such that each member shares a factor with either the first or the last member. In decimal, all powers of 76 also end in the same last two digits, the only other two-digit number with such property is 25. Karl A. Dahlke proved that 76 heptominoes of this shape, are able to fill a rectangle The atomic number of osmium, the Little Dumbbell Nebula in the constellation Pegasus is designated as Messier object 76. Seventy-six is also, In colloquial American parlance, reference to 1776, the Spirit of 76, patriotic painting by Archibald MacNeal Willard

76 (number)
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The Bennington Flag features the number 76

19.
81 (number)
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81 is the natural number following 80 and preceding 82. 81 is, the square of 9 and the power of 3. A perfect totient number like all powers of three, the ninth member of the Mian-Chowla sequence. A palindromic number in bases 8 and 26, a Harshad number in bases 2,3,4,7,9,10 and 13. One of three numbers which, when its digits are added together, produces a sum which. The inverse of 81 is 0.012345679 recurring, missing only the digit 8 from the set of digits. This is an example of the rule that, in base b,12 =0. 012 ⋯ ¯, omitting only the digit b−2, messier object M81, a magnitude 8. The duration of Saros series 81 was 1280.1 years, further, the number of lunar eclipse series which began on -0020 February 19 and ended on 1296 April 19. The duration of Saros series 81 was 1316.2 years, eighty-one is also, The number of squares on a shogi playing board The year AD81,81 BC, or 1981. The atomic number of thallium The symbolic number of the Hells Angels Motorcycle Club, H and A are the 8th and 1st letter of the alphabet, respectively. The code for international direct dial phone calls to Japan One of two ISBN Group Identifiers for books published in India Number of stanzas or chapters in the Tao te Ching, Number of prayers said in the Rosary in each night. The 81 is a 1965 song by Candy and the Kisses, artemis 81 is a 1981 BBC TV science fiction drama. The Eighty-One Brothers is a Japanese fable The most points NBA mega-star Kobe Bryant score in a game, the Arabic characters for the numerals 8 and 1 are visible in the left palm of the human hand. In China,81 always reminds people Peoples Liberation Army as it was founded on August 1,81 is used to refer to the motor-club Hells Angels, since H and A are, respectively, the 8th and 1st letters of the alphabet

81 (number)
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Department 81 of France (Tarn)

20.
98 (number)
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98 is the natural number following 97 and preceding 99. 98 is a Wedderburn-Etherington number and a nontotient, messier 98, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices. The New General Catalogue object NGC98, a magnitude 12.7 spiral galaxy in the constellation Phoenix.6 degrees Fahrenheit is normal body temperature 98, the number on a Ford Fusion in the NASCAR Sprint Cup Series, formerly driven by Josh Wise in 2015

98 (number)
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European route E98

21.
121 (number)
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121 is the natural number following 120 and preceding 122. One hundred twenty-one is a square and is the sum of three consecutive primes, there are no squares besides 121 known to be of the form 1 + p + p 2 + p 3 + p 4, where p is prime. Other such squares must have at least 35 digits, there are only two other squares known to be of the form n. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured that 4 and 121 are the perfect squares of the form x3 -4. It is also a number and a centered octagonal number. In base 10, it is a Smith number since its digits add up to the value as its factorization. But it can not be expressed as the sum of any other number plus that numbers digits, making 121 a self number

121 (number)
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A Chinese checkers board has 121 holes