Prove if m is irrational, then 5m is irrational. (Contraposition)

Prove if m is irrational, then 5m is irrational.

Remember that contraposition is where we have to prove P => Q so we use the following logic:

~Q => ~P so therefore P => Q

Thus we start off assuming the opposite of the consequent (~Q) which is that 5m is rational.

If that is true then 5m can be equal to two integers j and k(non-zero) such that 5m = j/k , since a rational number is one that can be expressed as the ratio of two integers, and an irrational number is one which can't, like 3.333333 or Pi.

So 5m = j/k and m = j/(5*k) , and thus m is rational, so now we have ~P for the contraposition part of the proof, and so we can say ~Q => ~P so therefore P => Q and so if m is irrational then 5m is irrational