# which is impossible.docx - which is impossible Thus it is...

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which is impossible. Thus it is apparent that these integer multiples of a are distinct and must be congruent modulo p to 1 , 2 , 3 ,...,p − 1. Multiplying these together we get, a · 2 a · 3 a ···( p − 1) a ≡ 1 · 2 · 3···( p − 1)(mod p ) a p −1 ( p − 1)! ≡ ( p − 1)!(mod p ) By cancelling ( p − 1)! from both sides we get the desired result. Example 2.12. Let p = 7 and a = 5 , then 7 - 5 and, 5 7−1 = 5 6 = 15625 and 15625 − 1 = (2232)(7) 15625 ≡ 1( mod 7) Similarly, for p = 13 and a = 28 , then 13 - 28 and, 28 13−1 = 28 12 = 232218265089212416 232218265089212416 − 1 = (17862943468400955)(13) 232218265089212416 ≡ 1( mod 13)
2.3 Euler’s Criterion and Legendre Symbol