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Q: 8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1…
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Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4
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Q: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False
A: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False
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For any odd prime p and integers a,b,c . If p doesn't divide ab then the congruence ak^2+bz^2 =c mod p has solutions
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- a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)In the congruences ax b (mod n) in Exercises 40-53, a and n may not be relatively prime. Use the results in Exercises 38 and 39 to determine whether these are solutions. If there are, find d incongruent solutions modulo n. 42x + 67 23 (mod 74)In the congruences in Exercises, and may not be relatively prime. Use the results in Exercises and to determine whether there are solutions. If there are, find incongruent solutions modulo. 50.