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: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

A:

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…

A:

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

A:

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

A:

Q: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

A:

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: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

A:

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

A:

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

A:

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: Take p and q are primes and p=3mod4 and q=3mod4. Take n=pq such that an integer z has a square root…

A: Given that р and q are primes and p=3mod4 and q=3mod4.

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: 3. Prove: for every integer n and for every odd integer a² = 1 (mod 2n+2)

A:

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

A: m is positive greater than 1.

Q: Prove Euler’s criterion, which states that if p is an odd prime and a is a positive integer not…

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

A:

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 a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

A: We have to find the moduler multiplicative inverse of a.

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…

A:

Q: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

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

A:

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…

A:

Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

A:

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

A:

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

A:

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

A:

Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

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

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: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

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: 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: 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: 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)

Q: Show that : If n is an integer then n is congruent to 0 modulo 4 or 1 modulo 4.

A:

Q: Let p be a prime such that p = 1 (mod 4). Prove that -lF, is a square in Fp.

A: