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Divisors, factors and multiples of integers; common divisor, greatest common divisor, common multiple, least common multiple, division algorithm, Euclid’s algorithm, unique factorization theorem

a divides b. We say an integer a divides an integer b (written "a | b") if there exists an integer c such that b = ac. When a | b we say that a is a factor of b, a is a divisor of b, or b is a multiple of a.

Example. 2 | 10 since 10 = 2·5 . The integer 10 is a multiple of 2.

Note the following: 3 divides 0 and, in general, a | 0 for all a ε I . Why? It follows as a direct consequence of the above definition since 0 = a·0 . This is a case of a result that would not be expected from the general concept of b as being a multiple of a, but which follows as a consequence of the axiomatic style definition.

Theorem 1. If a | b and a | c then a | (bx + cy) for all integers x,y.

Prime number. An integer p which is not 0 or +1 and is divisible by no integers except +1 and +p.

Common divisor of two or more quantities. A quantity which is a factor of
each of the quantities. A common divisor of 10, 15, and 75 is 5; a common divisor of x^{2} - y^{2}
and x^{2} - 2xy + y^{2} is x - y , since x^{2} - y^{2} = (x - y)(x + y) and x^{2} - 2xy + y^{2} = (x - y)^{2}.

Syn. common factor, greatest common measure.

Greatest common divisor (g.c.d.) of two or more quantities. A common divisor that is divisible by all other common divisors. For positive integers, the greatest common divisor is the largest of all common divisors e.g. the common divisors of 30 and 42 are 2, 3, and 6, the largest being the greatest common divisor 6. If we consider negative integers the common divisors of 24 and 60 are 1, 2, 3, 4, 6, and 12. and the greatest common divisors are 12. Syn. greatest common factor, greatest common measure.

The greatest common divisor of a and b is denoted by (a,b).

Common multiple of two or more quantities. A quantity which is a multiple
of each of two or more given quantities. The number 6 is a common multiple of 2 and 3. x^{2} - 1
is a common multiple of x - 1 and x + 1.

Least common multiple (l.c.m.) of two or more quantities. The least
quantity that is exactly divisible by each of the given quantities; 12 is the l.c.m. of 2, 3, 4, and 6.
The l.c.m. of a set of algebraic quantities is the product of all their distinct prime factors, each
taken the greatest number of times it occurs in any one of the quantities; the l.c.m. of x^{2} - 1 and
x^{2} - 2x + 1 is (x - 1)^{2}(x + 1) .

Tech. The l.c.m. of a set of quantities is a common multiple of the quantities which divides every common multiple of them.

Division Algorithm. For any integer a and any positive integer b, there exist unique integers q and r such that

a = bq + r, 0 r < b

The integer a is the dividend, b is the divisor, q is the quotient and r is the remainder. [Note. This theorem could be stated differently as “the quotient a/b equals q plus a remainder of r” – which explains the terminology.] For polynomials, the division algorithm states that, for any polynomial f and any non-constant polynomial g, there exist unique polynomials q and r such

where either r = 0 or the degree of r is less than the degree of g. The polynomials f, g, q, and r are the dividend, divisor, quotient, and remainder.

Euclid’s algorithm. A method of finding the greatest common divisor (g.c.d.) of two numbers – one number is divided by the other, then the second by the remainder, the first remainder by the second remainder, the second by the third, etc. When exact division is finally reached, the last divisor is the greatest common divisor of the given numbers (integers). In algebra, the same process can be applied to polynomials. E.g., to find the greatest common divisor of 12 and 20, we have 20 12 is 1 with remainder 8; 12 8 is 1 with remainder 4; and 8 4 = 2; hence 4 is the g.c.d.

Theorem 2. Any two integers a ≠ 0 and b ≠ 0 have a positive greatest common divisor (g.c.d.) which can be expressed as a “linear combination” of a and b in the form d = au + bv for integers u and v.

Theorem 3. If p is a prime, then p | ab implies p | a or p | b.

Theorem 4. If p is a prime and if p is a divisor of the product a∙b∙c∙ .... ∙t of n integers, then p is a divisor of at least one of these integers.

Relatively prime integers. Two integers a and b are said to be relatively prime if

(a,b) = 1.

The unique factorization Theorem. Every integer a > 1 has a unique factorization, except for order, into a product of primes

References.

1. James/James. Mathematics Dictionary.

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