Show that if a. b, k. and m are integers such that k ≥ 1, m ≥ 2, and a ≡ b (mod m), then ak ≡ bk(mod m).

Step 1</p>

In this problem we have to show that if a. b, k. and m are integers such that

k ≥ 1, m ≥ 2, and a ≡ b (mod m),

then ak ≡ bk(mod m).

Step 2</p>

ak ≡ bk(mod m) states that is congruent to bk(mod m).

Or, in other word we can say m divides integer and we get another integer, say quotient q.

In mathematical form we can write

When we expand then we get,

Let, which is also an integer.

we have

Therefore we can write ak ≡ bk(mod m) as

. ………………….(1)