Problem 9.1

Prove that for any integer n the product n(n + 1) is even.

Problem 9.2

Prove that the square of any integer has the form 4k or 4k + 1 for someinteger k

Problem 9.3

Prove that for any integer n, n(n2 1)(n + 2) is divisible by 4.

Theorem 9.2

Given any nonnegative integer n and a positive integer d there exist integers q and r such that n = dq + r and 0 r d. The number q is called the quotient of the division of n by d and we write q = n div d. The number r is called the remainder and we write r = n mod d or n r(mod d).