Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order 2n+¹ modulo p.

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

Q: Suppose that 2n−1 ≡ 1 (mod n). Is n necessarily prime?

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Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

Q: ) Find all possible values of m: 53 = -7 mod (m)

A: m is positive greater than 1.

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

Q: Let p be an odd prime and a, b E Z such that (a, p) 1 and (b, p) = 1. Prove that if neither x² = a…

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Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: 14. Prove that if p and q are distinct primes, then p9-1+q-1=1 (mod pq).

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Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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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: 2) Find the smallest positive integer z, such that z = 13 mod 4 z = 20 mod 9 z = 11 mod 25 .

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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: Show that (a + p)P = a (mod p2), where p is prime, and that, more generally, a = b (mod p) ⇒a= b…

A: To prove that (a+p)p≡ap(modp2), where p is a prime, it can use binomial expansion and Fermat's…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: 4. Let p> 2 be prime. Show that there exists an integer n such that n² + 1 is a multiple of p if and…

A: We use the fact that if gcd(a,p)=1 then a^{p-1}= 1 (mod p)