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

Q: Let n be a positive integer. Prove 41+3n mod 9.

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Q: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

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Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: Suppose a is an integer that is relatively prime to 400. Prove that a40 = 1 (mod 400).

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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: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

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Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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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: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

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

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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: Prove that if p is a prime and a is any integer so that p does not divide a, then the additive order…

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Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Prove that for every prime p, there exists an integer r, such that r = 16 mod p.

A: To prove: For every prime p, there exists an integer x, such that x8≡16 modp.

Q: For which positive values of k is nk = n (mod 7) for all integers n?

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

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Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: 13. a. Prove that for every integer n > 1, 10" = (-1)" (mod 11).

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Q: Suppose that m is a positive integer. Use mathematical induction to prove that if a and b are…

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

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Q: Find a number a such that 70a = 1 mod 17.

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Q: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

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Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

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: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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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: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

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: a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the…

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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…

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

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Q: Find a two digit integer c such that 101c = 1 mod 107. This congruence has a unique solution mod 101…

A: We need to find a digit integer c such that 101c ≡ 1 mod 107. As per the given question, this…

Q: Prove that for every odd prime 1000 < p < 2000 the number of –2, –1,2 which are squares mod p is not…

A: To prove that for every odd prime numbers strictly between 1000 and 2000 the numbers -2,-1, 2 which…

Q: Prove that for each integer a, if a not equal to 0 (mod 7), then a2 not equal to 0 (mod 7)

A: Consider the provided question,

Q: Let x be an integer. Prove that if x mod 3 is not 1, then 3|(x2+x).

A: Given x be an integer and xmod3≠1 then We have to prove that 3|x2+x Given xmod3≠1 This implies 3/x…

Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

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: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: The least positive integer x satisfying 22010 = 3x (mod 5) is

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Q: Prove that for every prime p, there exists an integer x, such that x = 16 mod p.

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