How do you prove the integers of the form 3n plus 1 is close under multiplication?

Answer 1

Suppose x and y are any two elements of the set.Then x = 3m + 1 and y = 3n + 1 for some integers m and n.

Then x*y = (3m + 1)*(3n + 1) = 9mn + 3m + 3n + 1 = 3*(3mn + m + n) + 1.

Since m and n are integers, mn+ m + n is also an integer = p, say.

Then x*y = 3p + 1.

That is the set is closed under multiplication.