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.

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

Thank you.

I think from $d|an$ you might just get $d|n$ and not $d|a.$ – coffeemath Dec 04 '19 at 07:39 — coffeemath, Dec 04 '19 at 07:39

score 0 · Answer 1 · answered Dec 04 '19 at 07:40

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Hint.

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

answered Dec 04 '19 at 07:40

Shubham Johri

score 0 · Accepted Answer · answered Dec 04 '19 at 15:49

0

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

answered Dec 04 '19 at 15:49

Bill Dubuque

sorry, but i'm a little confused at how you arrived to the conclusion above. could you show how you arrived from the universal property, a,b∣m⟺lcm(a,b)∣m to the above? – supremus_58 Dec 04 '19 at 22:25
@supremus_58 For $,m = ab,$ it says $,a,b\mid ab,\Rightarrow,{\rm lcm}(a,b)\mid ab,,$ i.e. the lcm divides every common multiple $,m,$ of $,a,b,,$ so in particular it divides the common multiple $,m = ab.\ $ – Bill Dubuque Dec 05 '19 at 05:39

2 Answers2