Prove that if n is a positive integer, then n is odd if andonly if 5n + 6 is odd.

3. Use induction to prove that 157+2 + 16²n+1 is divisible by 241 for every nonnegative integer n.

Prove by induction that (2n+1)3 is an odd number for all positive integers n.

Decide which of the following are valid proofs of the following statement: Let n be any integer. Then n³-n is divisible by 3. a) Assume n³ — n is divisible by 3. Let n³ n = 3b for some integer b. Let n = k + 1. Then n³ − n = (k + 1)³ – (k + 1) = k³ + 3k² + 3k + 1 − (k+ 1) = k³ + 3k² + 3k + 1 − k − 1 = (k³ − k) + 3k² + 3k = 3b + 3k² + 3k because k³ - k divisible by 3. = 3(b + k² + k) which is clearly divisible by 3. b) Note that n³ n = n(n² - 1) = n(n+1)(n − 1) which is the product of 3 consecutive integers, one of which must be a multiple of 3. Thus the product is a multiple of 3. c) Note that n³ - n = = n(n² — 1) and 3 is prime, so 3 | (n³ − n) if and only if either 3 | n or 3 | (n² — 1) (i.e. 3 divides one of its factors.) We have 3 cases depending on the remainder when n is divisible by 3 (also known as n mod 3). Let us address each case separately. case n = 0 case n = 1 mod 3. Then 3 n so 3 | n(n² — 1). mod 3. Then n = 3k + 1 for some integer k. Note that (3k + 1)² − 1 = 9k² + 6k+1…

If n is an integer, then (2n² + 3n) # 3(mod5).

Prove with 2k or 2k+1 and cases if necessary

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

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

Show that if 2n – 1 is a prime number, then n is a prime number.

Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even.