8. 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: 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: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

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

A:

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

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?

A:

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: Prove or disprove by using "congruence modulo n" and "divides." (a) For all positive integers a, x…

A:

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: 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).…

A:

Q: Exercise 16. Prove (without induction) that B" = 0 mod B for all positive integers B and n. Hint:…

A: Congruence and proof without induction

Q: Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order…

A:

Q: Compute the order of 2 with respect to the prime moduli 3, 5, 7, 11, 13, 17, and 19.

A:

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

A:

Q: Characterize the primes p mod 20 such ial

A: We have to characterize p mod 20 such that -5p=1.

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

A:

Q: 5. Prove that every integer is congruent modulo 9 to one of the even integers 0, 2, 4, 6, ..., 16.

A: To prove that every integer is congruent modulo 9 to one of the even integers

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

A:

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

A:

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: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a > 0.       We need to prove that a is even if and only if a ≡ 0…

Q: 6. Let m be an odd positive integer. Prove that every integer is congru- ent modulo m to one of the…

A:

Q: 1. Let m be a positive integer. If a = b (mod m) and c= d (mod m), a) Prove a+c= b + d (mod m) b)…

A:

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

A:

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)