# Math Help - Proof that m is divisible by 3 iff m^2 is divisible by 3

1. ## Proof that m is divisible by 3 iff m^2 is divisible by 3

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.

2. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

What you have done so far is correct (though you have used the wrong implies symbol, you should be using $\displaystyle \implies$), 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 $\displaystyle m = 3p + 1$ or $\displaystyle m = 3p + 2$ only.

If $\displaystyle m = 3p + 1$ then

\displaystyle \begin{align*}m^2 &= (3p + 1)^2 \\ &= 9p^2 + 6p + 1 \\ &= 3(3p^2 + 2p) + 1 \end{align*}

which is not divisible by 3.

Do the same for $\displaystyle m= 3p + 2$.

3. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

a direct proof of the reverse implication:

3 is prime, so if 3|ab, either 3 divides a or 3 divides b. now take a = b = m.

4. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

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

5. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

hi Duke.
there is a very general result which may help you in many situations which is:
If $p$ is a prime and if $p|a^n$ then $p|a$.

6. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

Originally Posted by Duke
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.

You can also see that fact with geometry:

7. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

Originally Posted by abhishekkgp
hi Duke.
there is a very general result which may help you in many situations which is:
If $p$ is a prime and if $p|a^n$ then $p|a$.

8. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

Originally Posted by Duke
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
That is Euclid's lemma - Wikipedia, the free encyclopedia

9. ## Re: Proof that m is divisible by 3 iff m^2 is divisible by 3

yes it is an "iff", but the "if" part is trivial. the "only if" part is sometimes used as the defintion of a prime number.