In mathematics, a divisor of an integer n called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one says that n is a multiple of m. An integer n is divisible by another integer m. If m and n are nonzero integers, more nonzero elements of an integral domain, it is said that m divides n, m is a divisor of n, or n is a multiple of m, this is written as m ∣ n, if there exists an integer k, or an element k of the integral domain, such that m k = n; this definition is sometimes extended to include zero. This does not add much to the theory, as 0 does not divide any other number, every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. In ring theory, an element a is called a "zero divisor" only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers. 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. 1 and −1 divide every integer. Every integer is a divisor of itself. Integers divisible by 2 are called and integers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n, not a trivial divisor is known as a non-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits. 7 is a divisor of 42 because 7 × 6 = 42, so we can say 7 ∣ 42. It can 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, −3. The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42; the set of all positive divisors of 60, A = ordered by divisibility, has the Hasse diagram: There are some elementary rules: If a ∣ b and b ∣ c a ∣ c, i.e. divisibility is a transitive relation.

If a ∣ b and b ∣ a a = b or a = − b. If a ∣ b and a ∣ c a ∣ holds, as does a ∣. However, if a ∣ b and c ∣ b ∣ b does not always hold. If a ∣ b c, gcd = 1 a ∣ c; this is called Euclid's lemma. If p is a prime number and p ∣ a b p ∣ a or p ∣ b. A positive divisor of n, different from n is called a proper divisor or an aliquot part of n. A number that does not evenly leaves a remainder is called an aliquant part of n. An integer n

Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, many mathematical proofs depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can be used in more advanced settings; the name is needed because there are operations, such as division and subtraction, that do not have it. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A corresponding property exists for binary relations; the commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a certain binary operation the two elements are said to commute under that operation; the term "commutative" is used in several related senses.

Putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same. In contrast, putting on underwear and trousers is not commutative; the commutativity of addition is observed. Regardless of the order the bills are handed over in, they always give the same total. Two well-known examples of commutative binary operations: The addition of real numbers is commutative, since y + z = z + y for all y, z ∈ R For example 4 + 5 = 5 + 4, since both expressions equal 9; the multiplication of real numbers is commutative, since y z = z y for all y, z ∈ R For example, 3 × 5 = 5 × 3, since both expressions equal 15. Some binary truth functions are commutative, since the truth tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p; this function is written as p IFF q, or as p ≡ q, or as Epq. The last form is an example of the most concise notation in the article on truth functions, which lists the sixteen possible binary truth functions of which eight are commutative: Vpq = Vqp.

Further examples of commutative binary operations include addition and multiplication of complex numbers and scalar multiplication of vectors, intersection and union of sets. Concatenation, the act of joining character strings together, is a noncommutative operation. For example,EA + T = EAT ≠ TEA = T + EAWashing and drying clothes resembles a noncommutative operation. Rotating a book 90° around a vertical axis 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order; the twists of the Rubik's Cube are noncommutative. This can be studied using group theory. Thought processes are noncommutative: A person asked a question and a question may give different answers to each question than a person asked first and because asking a question may change the person's state of mind; the act of dressing is either non-commutative, depending on the items. Putting on underwear and normal clothing is noncommutative. Putting on left and right socks is commutative.

Shuffling a deck of cards is non-commutative. Given two ways, A and B, of shuffling a deck of cards, doing A first and B is in general not the same as doing B first and A; some noncommutative binary operations: Division is noncommutative, since 1 ÷ 2 ≠ 2 ÷ 1. Subtraction is noncommutative, since 0 − 1 ≠ 1 − 0; however it is classified more as anti-commutative, since 0 − 1 = −. Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for = and = are Function composition of linear functions from the real numbers to the real numbers is always noncommutative. For example, let f = 2 x + 1 and g = 3 x + 7. = f = 2 + 1 = 6 x + 15 and ( g


The BMW M43 is an SOHC four-cylinder petrol engine, produced from 1991-2002. The M43 powered base-model cars, while higher performance models at the time were powered by the BMW M42 and BMW M44 DOHC engines; the M43 was produced at the Steyr engine plant. A version using natural-gas was produced for the E34 518i. Following the introduction of the BMW N42 engine in 2001, the M43 began to be phased out. Compared with its BMW M40 predecessor, the M43 features a timing chain, it features a dual length intake manifold, to provide torque across a wider rev range. In 1998 the displacement was increased to increasing torque to 180 N ⋅ m at 3900 rpm; the 1,596 cc M43B16 produces 75 kW and 150 N⋅m of torque. It uses the Bosch Motronic 1.7.1 engine management system. There was a natural gas-powered version of this car for the 1995 BMW 316g Compact. Applications: 1993-1999 E36 316i 1995-2000 E36/5 316g Compact 1998-2001 E46 316i The M43B18 has a 1,796 cc displacement, it uses the Bosch Motronic 1.7.1 fuel injection system.

There was a less powerful natural gas-powered version of this car for the BMW 518g Touring. This model was only available for two years. Applications: 1992-1998 E36 318i 1994-1996 E34 518i 1995-1996 E34 518g Touring 1995-2001 Z3 1.8 The M43B19 is the largest M43 engine, with a displacement of 1,895 cc. It produces up to 87 kW and 180 N⋅m, uses BMW's BMS 46 engine management system; the 77 kW versions do not have the DISA intake manifold. Applications— 77 kW and 165 N⋅m: 1999-2001 E36 316i 1999-2001 E46 316iApplications— 87 kW and 180 N⋅m: 1998-2001 E46 318i/318Ci 2001-2003 Z3 1.9 List of BMW engines