Show that the cube of a positive integer of the form \(6q + r\), where \(q\) is an integer and \(r = 0, 1, 2, 3, 4, 5\), is also of the form \(6m + r\).
Yes. The cube of such an integer is again of the form \(6m + r\).
Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
Exactly one out of n, n + 2 and n + 4 is divisible by 3.
Prove that one of any three consecutive positive integers must be divisible by 3.
Yes, in every set of three consecutive integers, one of them is divisible by 3.
For any positive integer \(n\), prove that \(n^3 - n\) is divisible by 6.
Yes. For every positive integer \(n\), the number \(n^3 - n\) is divisible by \(6\).
Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
Exactly one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5.