My professor proposed a question in class awhile ago and told us to prove it for homework. When he proved it in class the day it was due, he gave a very long and unnecessary proof. My proof was significantly shorter and when I asked my professor if my proof was correct he told me to go away, so, since I have such a helping professor, I have to know if this proof is correct so that I can study for the exam later on using the right information. Here's the statement:

If $\displaystyle a,b > 0$, then $\displaystyle (a,b) = (a + b, [a,b])$

Where (,) is the gcd and [,] is the lcm.

Proof:

Suppose there are two integers s and t s.t.

$\displaystyle p^s || a$ and $\displaystyle p^t || b$

This implies that:

$\displaystyle a = mp^s$

$\displaystyle b = np^t$

Where $\displaystyle p \not | mn$

Without loss of generality, assume $\displaystyle s \leq t$,

$\displaystyle a + b = mp^s + np^t$

$\displaystyle = p^s(m + np^{t -s})$

This implies that $\displaystyle p^s || (a + b)$

We know that $\displaystyle p^{max(s,t)} || [a,b]$ since $\displaystyle p^{max(s,t)}$ is the lcm by definition, and that $\displaystyle max(s,t) = t$

This implies that $\displaystyle p^t || [a,b]$

Therefore

$\displaystyle p^{min(s,t)} || (a + b, [a,b])$, where $\displaystyle p^s || a + b$ and $\displaystyle p^t || [a,b]$

and $\displaystyle p^{min(s,t)} || (a,b)$ by definition. Since they both are exactly divisible by the same power of p, they must be equal.

Thanks for the help.