The preceding part of the problem proves the reverse implication unless n = p2 where p is a prime. Prove that if p > 2 is prime then (p2− 1)! is congruent to 0 mod p2, and find k ∈ {0, 1, 2, 3} such that (22 − 1)! is congruent to k mod 4.

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: Prove by mathematical induction that for every positive integer n, 13 + 23 + 33 + .+ n³ n²(n+1)² is…

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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:

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Q: Use induction to prove that for every n E N, we have 5" = 1+ 4n (mod 8)

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Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: and drag on elements in order hese statements in order to produce a proof that the integers a and b…

A: Introduction: If the difference between integers a and b is a multiple of n, we say they are…

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: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: One number-theoretic result mentioned in the course was Wilson’s Theorem: If p is a prime then (p −…

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Q: Prove that 2" < n! for any integern 2 4

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p =&gt; αp-1…

Q: Prove that for any integers,s(s−1)(2s+ 5) is divisible by 6

A: Given that, s(s−1)(2s+ 5) we know that product of three consecutive integers is divisible by 6.…

Q: The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same…

A: Rewrite the following theorems using quantifiers and predicates.

Q: Prove that for any integer k, at least one of k, k + 2, or k + 4 is divisible by 3. (Hint: consider…

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Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

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Q: Establish that if a is an odd integer, then for any n ≥ 1 a 2 n ≡ 1 (mod 2n+2). [Hint: Proceed…

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Q: Is the following statement true or false? n+1 For every odd integer n, To answer this question, let…

A: Given statement true or fasle For every odd integer n {n /2} = n+1 /2

Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

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 the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: If a has order n-1 mod n, then n is prime. O True O False

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Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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Q: a) Prove that for all integers n > 0, 10" = 1(mod 9). You may refer to theorems from class or the…

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Q: Prove that if n is an odd integer, then n^2 is congruent to 1 modulo 8.

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Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

A: 41 = 4 42 = 16 43 = 64 44 =256   Conjecture:- For all n&gt;=0, If n is odd then Unit digit of 4n is…

Q: In the question that follows, the "Units Digit" or the "Unity Digit"of a number is the last digit in…

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

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Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

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Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

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Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

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Q: (a) Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: If a and b are integers and n is a positive integer, prove that a mod n =b mod n if and only if n…

A: If a and b are integers and n is a positive integer,    Formula used: a≡b mod n if and only if  a-b…

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not…

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Q: 2. Suppose that instead of proving the claim "there is no integer n such that n= 2 mod 6 and n7 mod…

A: consider an integer n=26 such that n mod 6=26 mod 6               =2 mod 6 also, n mod 9=26 mod 9…

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.