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This is an extremely simple problem but I'm new to this sort of math so I was wondering if anyone could lead me in the correct direction as to how I'd prove this formally?

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shea Kelse
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3 Answers3

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If $a\mid n$ and $b\mid n$, then $\operatorname{lcm}(a,b)\mid n$. Can you take it from there?

Ian Coley
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Since $\gcd(a,b)=1$, there are integers $x,y$ such that $ax+by=1$. Thus we can write$$\frac{n}{ab}=\frac{n(ax+by)}{ab}=\frac nbx+\frac nay,$$which is an integer since $\dfrac nb,\dfrac na,x,y$ are all integers.

bof
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Because $a,b$ are relatively prime, $\gcd(ab,n) = \gcd(a,n) \cdot \gcd(b,n)$

  • You have replaced the statement the poster wants to prove with a more general statement. Sometimes a more general statement is easier to prove. Is it so in this case? – bof Oct 09 '13 at 04:12
  • @bof: Different people learn and remember different facts; I felt it at least worth mentioning. –  Oct 09 '13 at 04:17