Consider the following statement. For every integer n, if n > 2 then there is a prime number p such that n 2, n! - 1 is an integer that is greater than 1. 4. So by the divisibility by a prime theorem (Theorem 4.4.4), there is a prime number p such that p|(n! – 1). . 5. Thus, p < n! - 1, and so p < n!. 6. Therefore, by steps 2 and 5, there is a prime number p such that n < p < n!.

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A:

Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: Prove that there is a natural number N such that for all natural numbers n > N, 2/n < 0.1.

A: Assume that there does not exist N such that for any natural number n,  n&gt;N, 2/n&lt;0.1.

Q: Prove that every prime p ≥ 5 is of the form p = √(24k + 1 )for some integer k.

A: To Prove that every prime p ≥ 5 is of the form p =√(24k + 1 )for some integer k. Or equivalently, p2…

Q: how many pairs of primes p and q are there such that q-p=3? Prove your answer.

A: Here, we need to find all the possible pairs of prime numbers p and q such that the difference (q-p)…

Q: A. Prove that for all positive natural numbers n, 13 + 23 + ... + n3 = n2⋅(n+1)2/4.

A: According to our guideline we can solve only first question of multiple question. If you want…

Q: 12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer,…

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Q: Prove the following statements: (a) If n is a positive integer then n 2 + 1 is an odd positive…

A: (a) n is an even positive integer. So, n = 2m for some integer m  So, n² + 1 = (2m)² + 1 = 2(2m²) +…