Use division algorithm to show that any positive odd integer is of the form 6q + 1, or 6q + 3 or 6q + 5, where q is some integer

Show that every positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Use division algorithm to show that the square of any positive integer is of the form 3p or 3p + 1.

Use division algorithm to show that the cube of any positive integer is of the form 9 m, 9m + 1 or 9m + 8.

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

Show that one and only one out of n, n + 2 or n + 4 is divisible by 3, where n is any positive integer

Find the sum ofthe first 40 positive integers divisible by 6.

Show that for any positive integer 3^(2n+2)-8n-9 is divisible by 64 .

Prove that the square of an odd integer can be expressed in the form 8p + 1 where p = 0 or a positive integer.

Show that, if n be any positive integer greater than 1, then (2^(3n) - 7n - 1) is divisible by 49.

Find q and r for the following pairs of positive integers a and b, satisfying a = bq + r. a = 125 , b = 5