This suggestion is in the form of a proof sorter activity: we’ve written out a proof of a certain property and then muddled the order, and your job is to put the various statements into the right order.

You might want to print and cut up the statements.

Claim: If \(p\) is prime and \(p\) divides \(ab\), then \(p\) divides \(a\) or \(p\) divides \(b\).

Note that ‘or’ in maths is inclusive, so you should read the conclusion above as “\(p\) divides \(a\) or \(p\) divides \(b\) or both”.

So \(p\) divides \(b\), as required.

There are integers \(m\) and \(n\) such that \(am + pn = 1\).

Assume that \(p\) is prime and that \(p\) divides \(ab\).

The left-hand side is a multiple of \(p\).

If \(p\) divides \(a\) then we are done, so assume that \(p\) does not divide \(a\). We want to show that \(p\) must divide \(b\).

So \(abm + pbn = b\).

Since \(p\) is prime, the highest common factor of \(a\) and \(p\) is \(1\).