Let a, b, and c be nonzero integers.

Question

Let a, b, and c be nonzero integers. Prove that if a divides b and a divides c, then a divides the difference b − c.

Answer 1

a divides b is given.

then a* n = b for some integer n.

a divides c is given.

then a * m = c for some integer m

subtracting the equations:

a * n - a* m = b - c

a(n - m) = b - c <-- distributive

since n and m are integers, n - m is

also an integer by closure property

of integer division.

So then there is an integer t = n-m

a* t = b - c

therefore,

t = (b-c)/a

which means a divides b - c

for integer t