if a prime p|LCM[a,b] then p|a or p|b

Question

Given a prime p show that if p|LCM[a,b] then p|a or p|b.

Is it enough for me to simply show that if d=LCM then d=an, then by Euclid(p|xy then p|x or p|y), p|d so p|an therefore p|a or p|n? and doing the same for b.

Also, how do I confirm the "or" here?

Thank you.

Answer 1

Hint: $\,\ p\mid {\rm lcm}(a,b)\mid ab\,\Rightarrow\, p\mid ab\:$ by the LCM Universal Property & transitivity of "divides".

Answer 2

Hint.

$$\text{LCM}(a,b)=a\cdot\frac b{\text{GCD}(a,b)}$$Now use Euclid's result.