Other Definitions
divisor (dict)
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Divisor - For divisors in algebraic geometry, see divisor (algebraic geometry).
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we usually write 7 | 42. Divisors can be positive or negative. The positive divisors of 42 are {1, 2, 3, 6, 7, 14, 21, 42}. Some special cases: 1 and -1 are divisors of every integer, and every integer is a divisor of 0. Numbers divisible by 2 are called even and those that are not are called odd. The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient. Rules for small divisors There are some rules which allow to recognize small divisors of a number from the number's decimal digits: A divisibility rule is a rule you can use to determine a number's divisibility by another number. In decimal, the divisibility rules are: - a number is divisible by 2 iff the last digit is divisible by 2
- a number is divisible by 3 iff the sum of its digits is divisible by 3
- a number is divisible by 4 iff the number given by the last two digits is divisible by 4
- a number is divisible by 5 iff the last digit is 0 or 5
- a number is divisible by 6 iff it is divisible by 2 and by 3
- a number is divisible by 7 iff the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2×4 = 28 is divisible by 7)
- a number is divisible by 8 iff the number given by the last three digits is divisible by 8
- a number is divisible by 9 iff the sum of its digits is divisible by 9
- a number is divisible by 10 iff the last digit is 0
- a number is divisible by 11 iff the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)
- a number is divisible by 12 iff it's divisible by 3 and by 4
- A number is divisible by 13
- iff the result of adding 4 times the last digit to the original number.
- iff the result of subtracting 9 times the last digit from the number with the last digit removed is divisible by 13 (e.g. 858 is divisible by 13 since 85-9×8 = 13 is divisible by 13).
- A number is divisible by 14 iff it's divisible by 2 and by 7.
- A number is divisible by 15 iff it's divisible by 3 and by 5.
Further notions and facts Some elementary rules: - If a | b and a | c, then a | (b + c).
- If a | b and b | c, then a | c.
- If a | b and b | a, then a = b or a = -b.
A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which 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. Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the Fundamental theorem of arithmetic. If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than that sum are said to be deficient, while numbers greater than that sum are said to be abundant. The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)). The relation | of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z. If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d. The rules for d=3 and d=9 given above are special cases of this result (b=10). Generalization One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting. See also External links
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