Number 11 (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 follov(a) If aP = bP (mod p), then a = b (mod p).(b) If aP = bP (mod p), then aP = bP (mod p²).[Hint: By (a), a = b+ pk for some k, so that aP -b = (b+ pk)P - b; now sthat 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) 1P +2P +3P + ..+ (p - 1)' = 0 (mod p).[Hint: Recall the identity 1+2 + 3+ .+ (p – 1) = p(p – 1)/2.112. Prove that if p is an odd prime and k is an integer satisfying 1

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(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

(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

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...

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