Let a and n be positive integers with a n and let m be the

Let a and n be positive integers with a < n, and let m be the smallest positive integer such that a^m is equivalent to a mod n. Prove that a^x is equivalent to a^n mod n if and only if x is equivalent to y mod m.

Solution

Since a and n are positive integers and a < n a mod n = a (Whwn a smaller number is divided by bigger number the remainder is smaller number)

As per the problem given

a mod n = a^m

So a^m = a

i.e. a^m = a¹

m = 1

y mod m = y mod 1 = 0 (any number divided by 1 gives remainder 0)

As given x = y mod m = 0

So x = 0

a^x = a^o = 1

n mod n = 0 (any number divided by same number gives remainder 0)

a^{n mod n} = a⁰ = 1

So a^x = a^{n mod n} if and only if x is equivalent to y mod m