Question

if a and b are odd numbers, is it closed under addition and multiplication? please explain why or why not?

Answer 1

No, adding two odd numbers gives you an even number.

Closed under multiplication because multiplying two odds gives you an odd.

Answer 2

This is fairly easy to prove: let \[a = 2n + 1\]\[b = 2m + 1\]then,\[a + b = 2n+1+2m+1 = 2(n+m+1)\]which is an even number since it's a multiple of two and\[a \cdot b = (2n + 1)(2m + 1) = 4mn + 2(n + m) + 1 = 2(2mn + n + m) + 1\]which is an odd number since it's not a multiple of 2 (it's a multiple of two plus one).