Use the Division Algorithm to prove that for all positive integers, n,6|(n(n + 1)(2n + 1)).

Use induction to prove that 16 divides 5n+2 + 13"+3 – 14 for all n > 0.

Find all positive integers n that is a product of two distinct primes such that o(n) = 60.

let M be the greatest integer less than 30 such that M!(M+1)!/2 is a perfect square. Let N be the greatest integer that divides C^4-C^2 for the integers C>1. Find M+N

Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

Use induction to prove that Σnk=1 (k + 3)/(k(k + 1)(k + 2)) = (5/4) - (2n+5)/(2(n+1)(n+2)) for all positive integers n.

Prove: For all positive integers a and b, if a^3 | b^2, then a | b. Hint: use prime power factorizations

Use the Division Algorithm to show that the cube of any integer is of the form 9k, 9k + 1 or 9k + 8. (Hint: By the Division Algorithm, the integer a is of one of the forms 9q, 9q + 1, ..., 9q + 8; establish the result for 9q + 3 and 9q + 7 only.)

Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Use the Principle of Mathematical Induction to prove that is true for all integers , where . Be sure to clearly state each part and circle your answer.