composite number

[logistic_guy]

Prove that if \(\displaystyle n\) is composite then there are integers \(\displaystyle a\) and \(\displaystyle b\) such that \(\displaystyle n\) divides \(\displaystyle ab\) but \(\displaystyle n\) does not divide either \(\displaystyle a\) or \(\displaystyle b\).

 

[logistic_guy]

If \(\displaystyle n\) is composite, then it can be expressed as the product of two integers greater than \(\displaystyle 1\), say \(\displaystyle n = ab\) where \(\displaystyle a > 1\) and \(\displaystyle b > 1\). Since both \(\displaystyle a\) and \(\displaystyle b\) are strictly less than \(\displaystyle n\), neither \(\displaystyle a\) nor \(\displaystyle b\) can be divisible by \(\displaystyle n\). However, their product \(\displaystyle ab=n\), so \(\displaystyle n \mid ab\).