Pembagi: Perbedaan antara revisi

* If <math>a \mid b</math> and <math>a \mid c</math>, then <math> a \mid (b + c)</math> holds, as does <math> a \mid (b - c)</math>.<ref><math>a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b+c=(j+k)a \Rightarrow a \mid (b+c)</math>. Similarly, <math>a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b-c=(j-k)a \Rightarrow a \mid (b-c)</math></ref> However, if <math>a \mid b</math> and <math>c \mid b</math>, then <math>(a + c) \mid b</math> does ''not'' always hold (e.g. <math>2\mid6</math> and <math>3 \mid 6</math> but 5 does not divide 6).

The total number of positive divisors of <math>n</math> is a [[multiplicative function]] <math>d(n)</math>, meaning that when two numbers <math>m</math> and <math>n</math> are [[relatively prime]], then <math>d(mn)=d(m)\times d(n)</math>. For instance, <math>d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7)</math>; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers <math>m</math> and <math>n</math> share a common divisor, then it might not be true that <math>d(mn)=d(m)\times d(n)</math>. The sum of the positive divisors of <math>n</math> is another multiplicative function <math>\sigma (n)</math> (e.g. <math>\sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42</math>). Both of these functions are examples of [[divisor function]]s.