For any integer a, a37 = ? mod 1729

Q: 1) Find an integer a and an integer y, such that x = 5 mod 10 y = 7 mod 12 x = 9 mod 21 y = 2 mod 17…

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Q: Prove that 01 + 02 is even.

A: Any odd integer, p can be represented as p=2k+1 . Any even integer, q can be represented as q=2k .…

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: (D) Let s be an integer such that 1 ss < 104651 and 24578° mod 104651 = 3190 Find s and explain how…

A: Let s be the integer such that and Purpose: To find the value of s

Q: For any integer a, prove that a = 0 or 1 (mod 3).

A: Every number is either of the forms 3k, 3k−1, 3k+1 where k is an integer

Q: (c) For each integer a, if a 4 (mod 8), then a 2 (mod 8).

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Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c…

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Q: Find all integers a such that = 6 mod 29. a. 21 (mod 29) O b. 1,0 (mod 29) O c. 10, 21 (mod 29) O d.…

A: According to Fermat's little theorem if p is any prime and a is any integer, then ap-1≡1modp here p…

Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer a,…

A: In this question, we have given assertion in terms of congruence modulo here we prove  8(a) if a is…

Q: a be an odd integer. Pro

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Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: Prove that for any integer k, at least one of k, k + 2, or k + 4 is divisible by 3. (Hint: consider…

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Q: For any integer a, prove that a=0 or 1 (mod 3).

A: The solution is given as follows

Q: Find an integer a such that a 400 = 1(mod 3911).

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Q: Find 134! (mod 713). Your answer should be an integer of the form all your steps.

A: Introduction: The modular expression is given as, 1341(mod 713). We have to find the value of the…

Q: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: Consider: for any integers a and b, if (a3 + b3) mod 3 does not equal 0, then (a + b) mod 3 does not…

A: The contrapositive of A implies B is not B implies not A. We use this and solve this problem.…

Q: If n| m and a = b(mod m), then a = b(mod n).

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Q: Suppose that a ≡ 9(mod 13) and b ≡ 7(mod 13). Find the integer cwith0≤c≤12suchthatc≡2a2 −6b2(mod13).

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Q: Q1 Prove or disprove i. If ac = bc(mod m), then a = b(mod m).

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Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

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Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: aibic, and m are integers such that IF 9,b,c, and m m70, dc, d|m, and ac = bc (mod m), then a =b…

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Q: Suppose that each of a, b, and c is an integer and m is a positive integer. Show that if a = b(mod…

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Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

Q: If a is an integer such that: 2 ła, 5ła Show that (a)⁴ⁿ = 1(mod 40) for any positive integer n

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Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: For every positive integer a, find a composite number n such that n|a" – a.

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Q: Let n be an integer. Prove that, if n mod6 = 2, then n2mod12=4

A: We apply concept of modulus here to solve this.

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Find the nonnegative integer a less than 28 represented by each of these pairs, where each pair…

A: According to the given information, it is required to find the non-negative integer a less than 28…

Q: Let each of a and m be an integer with m > 1. We know that if a and m are relatively prime, then a…

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Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

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Q: Let a E Z. Prove that if a = 2 (mod 6), then a? = 4 (mod 12).

A: We will prove this by basic result of congruence

Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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Q: If a and b are integers such that a? = b² (mod p), where p is prime integer, then a) a = b (mod p)…

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Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: If a = 2846463 mod 59, then a =? A

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Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.