Let a b belongs to N Prove that there exist k l belongs to N

Let a, b belongs to N. Prove that there exist k, l belongs to N such that k not equal to l and a | b^k - b^l.

Solution

Let a.b € N, where N= 1,2, 3, ........

There exists k,l € N s.t. k not l

Again let us define b^k = I₁ a and b^l = I₂ a, where I₁, I₂ € N

So, b^k - b^l = I₁a -l₂a = a(I₁ -l₂ )

Implies that a divides b^k - b^l for k,l € N.