Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod p. Show that then there is k E Z with k² + b•k +c = 0 mod p.
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A: The given question is about number theory. The solution is given below.
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A: Solution
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A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…
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A: The solution follows from the Generalized Euclidean Algorithm for finding GCD.
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Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).
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A: We prove the result by the method of contradiction.
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Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)
A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…
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A: The detailed solution is as follows below:
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Q: 1. Use Euler's Theorem to prove a = a (mod 105) for all a E Z.
A: Since you have asked multiple question, we will solve the first question for you. If you want any…
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A: ap = a (mod q) and aq = a (mod p)
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A: We will prove this by basic result of congruence
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A: The detailed solution is as follows below:
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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)True or False Label each of the following statements as either true or false. 2. and imply for .
- 29. Find the least positive integer that is congruent to the given sum, product, or power. a. b. c. d. e. f. g. h. i. j. k. l.Label each of the following statements as either true or false. The notation mod is used to indicate the unique integer in the range such that is a multiple of.True or False Label each of the following statements as either true or false. 1. implies for .