(b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29). 10. Assuming that a and b are integers not divisible by the prime p, establish the follow (a) If a' = bP (mod p), then a = b (mod p). (b) If aP [Hint: By (a), a = b+ pk for some k, so that aP -bP = (b+ pk)P – bP; now s that p2 divides the latter expression.] 11. Employ Fermat's theorem to prove that, if p is an odd prime, then (a) 1P-1+ 2P-l+3P-1+... + (p – 1)P- = -1 (mod p). (b) 1° + 2P + 3P + ..+ (p – 1)' = 0 (mod p). [Hint: Recall the identity 1+ 2 +3++ (p - 1) = p(p – 1)/2.] 12. Prove that if p is an odd prime and k is an integer satisfying 1 < k sp-1, then binomial coefficient = bP (mod p), then aP = bP (mod p2). = (-1) (mod p) d(a, pq) = 13. Assume that show that a9 14. If p and q ar Zoom out rton- Elementary..., McGraw-Hill 5th edition).pdf O 209% O 109 of 425 ET(mod pq) anaar %23