Let N be an integer, and let p be a prime > 7. Suppose p| (N2+7). Show that p is of the form 1+ 7k, 2 + 7k, or 4+7k.

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: Let p25 be a prime. Prove that p(p+1) (p-1)! +p+1.

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Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

A: We Shall solve one question only un per the company guidelines as you have asked more than one Que.…

Q: For any distinct primes p, q,r and s, we have pq # rs. For any positive integer n, there exists n…

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Q: Let n be an integer with n ≥ 2. Suppose that for every prime               p ≤√n, p does not divide…

A: given,                                                  p≤n p doesn't divide n. let n is not a prime…

Q: .If p 5 is an odd prime, prove that either p? – 1 or p² + 1 is divisible by 10.

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Q: Let a, b be integers for which a01 and ba02 are relatively prime. Show that a and b are also…

A: Given: a and b are two integers for which a101 and b202 are relatively prime. We have to show that a…

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

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Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

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Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: How to solve this?

A: According to the given information, let p be a prime number such that p>2. It is required to…

Q: P(n): 2"+1 > 2n + 1

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Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: Let n be any integer. If n=3k+5 for some integer k. Does 91 6n-3 No, because 6n-3= 18k+30-3 which is…

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Q: For all n E N,n > 1, there exists a prime p such that p|n. صواب İhi O

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Q: prove that there exists a prime number p such that n <p< 2n for n = 10,11,12,13,14, 15, 16, 20.

A: for n=10 then take p=19, 10&lt;=p&lt;=20  for n=11 then take p=19, 11&lt;=p&lt;=22 for n=12 then…

Q: For integer values of a, use a proof by contraposition (also called contrapositive or indirect…

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Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

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Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c &amp; p,…

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

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Q: Let pn be nth prime. prove that the sum 1/p1+1/p2+…..+1/pn is never an integer.

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Q: Prove: in Z12. L12, for every prime number p>iz, [p] = [1], [5], [7) , or [11]

A: Consider the set ℤ12 consist of all equivalence classes of integers obtained by the remainders…

Q: Give an example of a prime p of each of the forms: 6k + 1 6k + 5.

A: We need to give an example of a prime p of each of the given forms. Let us take k=1. So, we get…

Q: 11. If n is an odd integer, show that n + 4n² + 11 is of the form 16k.

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Show that if 2m +1 is an odd prime, then m 2" for some nonnegative integer n.

A: We solve this question by contradiction

Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: For every prime p, show that Х-1 — 1 %3 (х — 1)(х — 2) : .. [x — (р — 1)] in Zp[x].

A: Fermat's Little Theorem: Let p be a prime and let a∈N , 0&lt;a&lt;p. Let Ap={1a,2a,....(p-1)a} No…

Q: Prove that for all primes p that if p does not divide n2, then p does not divide n.

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Q: prove n2 + 5 greater than or equal to 4n for all integers n

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Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

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Q: Prove that if n = p1a1...prar, where the pi are prime, then σ(n)ϕ(n) = n2(1-p1-a,-1)...(1-pr-a,-1

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Q: For every prime p, show that XP-1 – 1 = (x – 1)(x – 2) · · · [x – (p – 1)] in Z„[x].

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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Q: Let pand g n=p8ge, be distinct primes. het show that v (^)n.

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Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

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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: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…