1.
Gregorian calendar
–
The Gregorian calendar is internationally the most widely used civil calendar. It is named after Pope Gregory XIII, who introduced it in October 1582, the calendar was a refinement to the Julian calendar involving a 0. 002% correction in the length of the year. The motivation for the reform was to stop the drift of the calendar with respect to the equinoxes and solstices—particularly the northern vernal equinox, transition to the Gregorian calendar would restore the holiday to the time of the year in which it was celebrated when introduced by the early Church. The reform was adopted initially by the Catholic countries of Europe, the last European country to adopt the reform was Greece, in 1923. Many countries that have used the Islamic and other religious calendars have come to adopt this calendar for civil purposes. The reform was a modification of a made by Aloysius Lilius. His proposal included reducing the number of years in four centuries from 100 to 97. Lilius also produced an original and practical scheme for adjusting the epacts of the moon when calculating the date of Easter. For example, the years 1700,1800, and 1900 are not leap years, but the years 1600 and 2000 are. The canonical Easter tables were devised at the end of the third century, when the vernal equinox fell either on 20 March or 21 March depending on the years position in the leap year cycle. As the rule was that the full moon preceding Easter was not to precede the equinox, the date was fixed at 21 March for computational purposes, the Gregorian calendar reproduced these conditions by removing ten days. To unambiguously specify a date, dual dating or Old Style, dual dating gives two consecutive years for a given date, because of differences in the starting date of the year, and/or to give both the Julian and the Gregorian dates. The Gregorian calendar continued to use the calendar era, which counts years from the traditional date of the nativity. This year-numbering system, also known as Dionysian era or Common Era, is the predominant international standard today, the Gregorian calendar is a solar calendar. A regular Gregorian year consists of 365 days, but as in the Julian calendar, in a leap year, in the Julian calendar a leap year occurs every 4 years, but the Gregorian calendar omits 3 leap days every 400 years. In the Julian calendar, this day was inserted by doubling 24 February. In the modern period, it has become customary to number the days from the beginning of the month, some churches, notably the Roman Catholic Church, delay February festivals after the 23rd by one day in leap years. Gregorian years are identified by consecutive year numbers, the cycles repeat completely every 146,097 days, which equals 400 years

2.
Divisor
–
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number

3.
Leap year
–
A leap year is a calendar year containing one additional day added to keep the calendar year synchronized with the astronomical or seasonal year. By inserting an additional day or month into the year, the drift can be corrected, a year that is not a leap year is called a common year. For example, in the Gregorian calendar, each year has 366 days instead of the usual 365. In the Bahai Calendar, a day is added when needed to ensure that the following year begins on the vernal equinox. For example, Christmas Day fell on a Tuesday in 2001, Wednesday in 2002, the length of a day is also occasionally changed by the insertion of leap seconds into Coordinated Universal Time, owing to the variability of Earths rotational period. Unlike leap days, leap seconds are not introduced on a regular schedule, in the Gregorian calendar, the standard calendar in most of the world, most years that are multiples of 4 are leap years. In each leap year, the month of February has 29 days instead of 28, adding an extra day to the calendar every four years compensates for the fact that a period of 365 days is shorter than a tropical year by almost 6 hours. Some exceptions to this rule are required since the duration of a tropical year is slightly less than 365.25 days. For example, the years 1700,1800, and 1900 were not leap years, over a period of four centuries, the accumulated error of adding a leap day every four years amounts to about three extra days. The Gregorian calendar therefore removes three leap days every 400 years, which is the length of its leap cycle and this is done by removing February 29 in the three century years that cannot be exactly divided by 400. The years 1600,2000 and 2400 are leap years, while 1700,1800,1900,2100,2200 and 2300 are common years, by this rule, the average number of days per year is 365 + 1⁄4 − 1⁄100 + 1⁄400 =365.2425. The rule can be applied to years before the Gregorian reform, the Gregorian calendar was designed to keep the vernal equinox on or close to March 21, so that the date of Easter remains close to the vernal equinox. The Accuracy section of the Gregorian calendar article discusses how well the Gregorian calendar achieves this design goal, the following pseudocode determines whether a year is a leap year or a common year in the Gregorian calendar. The year variable being tested is the representing the number of the year in the Gregorian calendar. Care should be taken in translating mathematical integer divisibility into specific programming languages, if then else if then else if then else February 29 is a date that usually occurs every four years, and is called leap day. This day is added to the calendar in leap years as a corrective measure, the Gregorian calendar is a modification of the Julian calendar first used by the Romans. The Roman calendar originated as a calendar and named many of its days after the syzygies of the moon, the new moon. The Nonae or nones was not the first quarter moon but was exactly one nundina or Roman market week of nine days before the ides and this is what we would call a period of eight days

4.
Intercalation (timekeeping)
–
Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months, the solar or tropical year does not have a whole number of days, but a calendar year must have a whole number of days. The most common way to reconcile the two is to vary the number of days in the calendar year. In solar calendars, this is done by adding to a year of 365 days. In the Julian calendar, as well as in the Gregorian calendar, in the Julian calendar this was done every four years. In the Gregorian, years divisible by 100 but not 400 were exempted in order to improve accuracy, thus,2000 was a leap year,1700,1800, and 1900 were not. Epagomenal days are days within a calendar that are outside any regular month. Usually five epagomenal days are included within every year, but a sixth day is intercalated every four years in some. The Baháí calendar includes enough epagomenal days before the last month to ensure that the year starts on the March equinox. These are known as the Ayyám-i-Há, the solar year does not have a whole number of lunar months, so a lunisolar calendar must have a variable number of months in a year. Regular years have 12 months, but embolismic years insert a 13th intercalary or embolismic month every second or third year, whether to insert an intercalary month in a given year may be determined using regular cycles such as the 19-year Metonic cycle or using calculations of lunar phases. The Buddhist calendar adds both a day and month on a usually regular cycle. Some historians also linked the practice of Nasi to intercalation. The International Earth Rotation and Reference Systems Service can insert or remove leap seconds from the last day of any month and these are sometimes described as intercalary. ISO8601 includes a specification for a 52-week year, any year that has 53 Thursdays has 53 weeks, this extra week may be regarded as intercalary

5.
February 29
–
February 29, also known as leap day or leap year day, is a date added to most years that are divisible by 4, such as 2008,2012,2016,2020, and 2024. A leap day is added in various solar calendars, including the Gregorian calendar standard in most of the world, lunisolar calendars instead add a leap or intercalary month. In the Gregorian calendar, years that are divisible by 100, thus,1700,1800, and 1900 did not contain a leap day, neither will 2100,2200, and 2300. Conversely,1600 and 2000 did and 2400 will, Years containing a leap day are called leap years. Years not containing a leap day are called common years, February 29 is the 60th day of the Gregorian calendar, in such a year, with 306 days remaining until the end of the year. In the Chinese calendar, this day will only occur in years of the monkey, dragon, a leap day is observed because a complete revolution around the Sun takes approximately 6 hours longer than 365 days. A leap day compensates for this lag, realigning the calendar with the Earths position in the Solar System, otherwise, although most modern calendar years have 365 days, a complete revolution around the Sun takes approximately 365 days and 6 hours. An extra 24 hours thus accumulates every four years, requiring that a calendar day be added to align the calendar with the Suns apparent position. A solar year is slightly shorter than 365 days and 6 hours. The currently accepted figure is 365 days,5 hours,48 minutes,45 seconds. Adding a calendar day every four years, therefore, results in an excess of around 44 minutes for those four years, to compensate for this, three days are removed every 400 years. The Gregorian calendar reform implements this adjustment by making an exception to the rule that there is a leap year every four years. Instead, a year divisible by 100 is not a year unless that year is also divisible by 400. This means that the years 1600,2000, and 2400 are leap years, while the years 1700,1800,1900,2100,2200,2300, the Gregorian calendar repeats itself every 400 years, which is exactly 20,871 weeks including 97 leap days. Over this period, February 29 falls on Sunday, Tuesday, and Thursday 13 times each,14 times each on Friday and Saturday, the calendar of the Roman king Numa Pompilius had only 355 days which meant that it would quickly become unsynchronized with the solar year. An earlier Roman solution to this problem was to lengthen the calendar periodically by adding days to February. February consisted of two parts, each with an odd number of days, the first part ended with the Terminalia on the 23rd, which was considered the end of the religious year, and the five remaining days formed the second part. To keep the calendar year roughly aligned with the solar year, the second part of February was incorporated in the intercalary month as its last five days, with no change either in their dates or the festivals observed on them

6.
Julian calendar
–
The Julian calendar, proposed by Julius Caesar in 46 BC, was a reform of the Roman calendar. It took effect on 1 January 45 BC, by edict, the Julian calendar gains against the mean tropical year at the rate of one day in 128 years. For the Gregorian the figure is one day in 3,030 years, the difference in the average length of the year between Julian and Gregorian is 0. 002%. The Julian calendar has a year of 365 days divided into 12 months. A leap day is added to February every four years, the Julian year is, therefore, on average 365.25 days long. It was intended to approximate the tropical year, as a result, the calendar year gains about three days every four centuries compared to observed equinox times and the seasons. This discrepancy was corrected by the Gregorian reform of 1582, consequently, the Julian calendar is currently 13 days behind the Gregorian calendar. Egypt converted on 20 December 1874/1 January 1875, turkey switched on 16 February/1 March 1917. Russia changed on 1/14 February 1918, Greece made the change for civil purposes on 16 February/1 March 1923 - the national day, which was a religious holiday, was to remain on the old calendar. Most Christian denominations in the west and areas evangelised by western churches have replaced the Julian calendar with the Gregorian as the basis for their liturgical calendars. However, most branches of the Eastern Orthodox Church still use the Julian calendar for calculating the date of Easter, some Orthodox churches have adopted the Revised Julian calendar for the observance of fixed feasts, while other Orthodox churches retain the Julian calendar for all purposes. The Julian calendar is used by the Berbers of the Maghreb in the form of the Berber calendar. In the form of the Alexandrian calendar, it is the basis for the Ethiopian calendar, during the changeover between calendars and for some time afterwards, dual dating was used in documents and gave the date according to both systems. In contemporary as well as texts that describe events during the period of change. The ordinary year in the previous Roman calendar consisted of 12 months, in addition, a 27- or 28-day intercalary month, the Mensis Intercalaris, was sometimes inserted between February and March. The net effect was to add 22 or 23 days to the year, some say the mensis intercalaris always had 27 days and began on either the first or the second day after the Terminalia. According to the later writers Censorinus and Macrobius, the ideal intercalary cycle consisted of ordinary years of 355 days alternating with intercalary years, alternately 377 and 378 days long. In this system, the average Roman year would have had 366 1⁄4 days over four years, Macrobius describes a further refinement whereby, in one 8-year period within a 24-year cycle, there were only three intercalary years, each of 377 days