Let m=3k, where k is a positive integer.
<-> m^2=9k^2
<-> m^2=3(3k^2)
<-> m^2 is divisible by 3.
That is my attempt but I don't think the last line is correct.
What you have done so far is correct (though you have used the wrong implies symbol, you should be using ), but you have only proven the if statement. To prove the iff statement (if and only if) you need to prove it in the other direction as well. I.e. show that if m^2 is divisible by 3, then so is m.
This will be easiest to prove using the contrapositive - i.e. if m is not divisible by 3, then neither is m^2.
This means that or only.
If then
which is not divisible by 3.
Do the same for .
Thanks for your very helpful answers. Couple more questions (1) When we talk about m being divisible by k, do we only consider m,k positive integers? (2) Deveno, is that an iff statement. 3|ab iff 3 divides a or b
If so surely we can do it in on go. 3|m^2 <-> 3 divides m or m <->3 divides m. Thanks