Let p and q be two distinct primes. (a) Prove that if a = b (mod p) and a = b (mod q), then a = b (mod pq) for any a,b € Z. (b) Use (a) and Fermat's Little Theorem to prove that p1+ 1 (mod pq).

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

Q: Let p be an odd prime. Prove that (a) If p = 1 (mod 5) or p = 4 (mod 5), then 5 is quadratic residue…

A: Given p is an odd prime. We have to prove (a). if p≡1(mod 5)  or  p≡4(mod 5), then 5 is a…

Q: Find the value of 592 (mod 11) using Fermat's Little Theorem. Remember, x (P-1) = 1 (mod p) a 592 =…

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Q: Let be P odd an prime s.t- gcd (y² + 4v₁ p) = 1. 2 If (x + y √B ) ² = u ² + 4v (mod pk) (VK 6 N),…

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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: The remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 5. (So a mod 7…

A: Given that remainder when "a" is divided by 7 is 2

Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

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Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: Let p be prime. Suppose a, b ∈ ℤ and ab congruent to 0 (mod p). Prove that either a congruent to 0…

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Q: Prove that's is prime p, then (a + b)² = (a² + b²)(mod p), Va; b € Z

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Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: 4. Let a, b, c, d, n € Z such that a = b (mod n) and c= d (mod n). (a) Prove that a+c=b+d (mod n).…

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Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: Prove that if x² = y² (mod n) and x #±y (mod n) then gcd(x + y, n) is a nontrivial factor of n.

A: We have,x2≡y2mod nAnd,x≢±ymod n. We have to show that gcdx+y, n is a nontrivial factor of n.

Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

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: (b) Wilson's Theorem: Let p be prime. Then (p-1)!= -1(mod p). Hint: Verify for p = 2 and p = 3…

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Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

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Q: Let p and q be distinct odd primes such that gcd(p−1, q−1) = 2. Does there exist an integer M such…

A: The objective of the question is to find out if there exists an integer M that satisfies the given…

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: 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: Let p be an odd prime, let a be an integer that is not divisible by p, and let b be a square root of…

A: We present the ideas involved in the problem. Basic Definition of quadratic residues and all will be…

Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: Problem 2. Let p be an odd prime number and let r be a primitive root (mod p). Thus, płr, rP-1 = 1…

A: Let p be an odd prime and r be a primitive root modulo p. ... Can Write A = R^k For Some K.  Prove…

Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: Consider the primes p = 5 and q = 11. Let e = 3. (a) Compute pq and (p − 1)(q − 1). Prove that…

A: First, we need to compute  and . Given  and we can calculate  as follows: Now, we need to prove that…

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: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: Prove that for any odd prime p and a, b = Z, ap+bP = 0 (mod p) ⇒ ap+bP = 0 (mod p²).

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Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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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: Ulia primeive Toute 95. Find all primes p ≤ 29 such that 2 is a primitive root modulo p.

A: Here we have to find all primes less than 29 or equal to 29 having primitive root 2.

Q: Let a, b and c a=1b=0C=9 example with enrolment number 001054321, you would take a = 0, b = 5 and c…

A: The answer to the given question is provided below with respective explanation.

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: Exercise 36 Prove that x? = a (mod p) has either no solutions or 2 solutions when p is a prime and a…

A: I have use the method of contradiction

Q: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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Q: a is any integer and p is prime, then prove that a\power{p}≡a (mod p

A: If a is any integer and p is prime, then we have to show that

Q: do I prove this: Let a, b, y ∈ Z and d ∈ N so that b ≡ y(mod d). Then a ⋅ b ≡ a ⋅ y(mod d).

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

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