3) Support the p is a that Show least one Prove then iss prime 27. ท (1^2) = (2+1) = 1 for at number {1, 2, ---,9} مروز n in the set that if p is طر an odd prime x² = 2 (mod p) p=1 or 7 (mod 8). has solutions

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Let p be a prime. What is the remainder of (p−1)!, when divided by p? In other words: find a simple…

A: Remainder will be (p-1). As p is a prime number. p is divisible by either p or 1. It will never get…

Q: Is there a solution to x^2 - 9x + 17 = 0 mod 73?

A: The given in this question is a quadratic equation in the form of x2-9x+17=0, and the objective is…

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: 7) Let p be an odd prime, find if a) p =1 mod 8

A:

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: 2) induction to that for all Use prove nonnes atine integers 1, 51 (n=-n)

A:

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: (()) b) Use part a) to find an integer that satisfies the congruence x² = -1 (mod 17) a) Prove that…

A:

Q: L-) Find all in tegers such thet 5x = 12 (mod 7) 10x = 4 (mod 11) E 16 (mod 10d ond 1occ < x < 3000

A: To determine: All the integers x such that 5x≡12 (mod 7)10x≡4(mod11)7x≡16(mod 10)and 1000≤x≤3000

Q: X=2 (mod 10), X=2 (Mod 8), x=o (Mod 6) What is the smallest positive number ?

A:

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

A:

Q: 1. Solve the Congruence x^5 ≡ 2 (mod 7)   2. Solve the congruence x^7 ≡ 10 (mod 23)   Show work…

A:

Q: 2221 (mod 13) is equal to O 10 (mod 13) O 1 (mod 13) 7 (mod 13) 6 (mod 13) O None of these

A:

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

A:

Q: gcd (k, Pal)=- x* =b (mod P). Theorm: If p aned q are dis tinbed primes non of which divides b, and…

A:

Q: Let m be an element of N, be a relatively prime to 5. Show that 5|(m^2 + 1)(m^2 + 4)

A:

Q: Show that II (1-) < II p° perfect = n = p*||n p*||n Hint: The following formula should prove useful:…

A:

Q: if p is any prime and a is any integer, then p/(a^p+(p-1)!a) and p/(a+(p-1)!a^p) Prove equivalent…

A: To prove the equivalence of the two statements, we need to show that if the first statement is true,…

Q: Determine the number of primitive roots modulo 23.

A: Solution

Q: Suppose that p is a prime number. Given this information prove that if x is an integer such that x^2…

A:

Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

A:

Q: Let me Z+ be odd with prime-power factorization m = p₁¹p2²p3³p4, where a1a2 a3 a4 P₁ = P2 = P3 = ±5…

A: As per the question we are given a odd numbers m and its prime power factorization as m =…

Q: Prove that there are infinity many primes . Please provide a typed solution

A: We have to prove that there are infinitely many primes. We will prove that by contradiction,…

Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

Q: Let p be a prime. Find the least residue mod p of 1^p + 2^p + ... + (p-1)^p

A: The given problem is to find the least residue mod p of 1p+2p+....+(p-1)p.

Q: (*:') • (":") - () n+1` + 1 a 3 2

A: From ncr formula we can easily derive the answer.

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Compute all least residue solutions modulo 12 of 8x-4 congruent to 0(mod 12)

A:

Q: Prove that there is an even integer x which is the sum of the cubes of 4 odd integers. Recall that…

A: We have 4 odd integers and and even integer x , and we need to prove the sum of cubes of 4 odd…

Q: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: てheorem' Let p be aprime number, a integer (ap)-1 0nd pacitive integer. In this cose; ) #1 (mod p)…

A: We need to use the above given two theorems to find the solution for the given congruence

Q: Find the smallest positive integer k satisfying the stated condition or explain no such k exists:…

A: To find the smallest positive integer k satisfying the given condition of congruence.

Q: 30. Find the integer a such that a) a = 43 (mod 23) and -22 < a s0. b) a = 17 (mod 29) and -14 < a s…

A:

Q: 5 how that Cm d n+2 Che prine 4C(n-1)! +)+n = 0 mod @ncnt2)

A: We show that the n and n+2 are primes If and only if  4((n-1)!+1)+n≡0 mod n(n+2) We take n as prime…

Q: Prove that 52-1 is divisible by 8 for all NEN 1) show for N=1 5263-1-N 52-1-21 =3 2(1) 24 8 8…

A: