a. Show that for every non-negative integers m, n, N = 7.5" has exactly (m + 1)(n+1) different positive divisors.

Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: 1. Prove that nº + n is even for every integer n.

A:

Q: n(nº +5) is divisible by 6, for each integer n 0.

A:

Q: 9. Prove that n - n is divisible by 30 for every integer n. te

A:

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

A:

Q: exp(ix())= 4k+1 2

A:

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

A:

Q: Show that for every positive

A:

Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: For every integer n > 0, 1+2+22 + · + 2" = 2"+1 – 1.

A: Given: The statement 1+2+22+⋯+2n=2n+1-1 Proof: Base step: Show the statement is true when n=0.…

Q: For all integers n> 0, 32n – 1 is divisible by 8.

A:

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

A:

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: 10. Prove that n³-n is divisible by 6 for every integer n.

A:

Q: Prove that 9? −1 is divisible by 8,for all n ≥ 1

A:

Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.