Fix a natural number n and let a,b,c,d be integers. Please prove that n divides (a-b) and n divides (c-d) then n divides (ac-bd)

Answer 1

n divides a-b means that a-b = kn, for some integer k.

n divides c-d means that c-d = rn, for some integer r.

So, ac = (b+kn)(d+rn)

ac = bd +rnb + knd + krn²

ac - bd = n(rb + kd + krn)

Since r,b,k,d, and n are integers, so is rb + kd + krn.

Therefore, ac - bd is divisible by n.