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2^a7^b=2^c7^d (mod 15) iff 3^a5^b =3^c5^d (mod 16)

Source: 1999 Estonia National Olympiad Final Round grade 12 p1

March 11, 2020
number theoryremainderpower of 2power of 3

Problem Statement

Let a,b,ca, b, c and dd be non-negative integers. Prove that the numbers 2a7b2^a7^b and 2c7d2^c7^d give the same remainder when divided by 1515 iff the numbers 3a5b3^a5^b and 3c5d3^c5^d give the same remainder when divided by 1616.
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