Prove gcf times the lcm of any two numbers equals the products of those two numbers

Given two numbers, l and m, prove that:

gcf(l, m)lcm(l, m) = lm

#NumberTheory #HelpMe! #Proofs #MathProblem #Math

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

Assuming that the prime factorisations of $l$ and $m$ are as follow,

$l=2^{a_1}\times3^{a_2}\times...\times P^{a_n}$

$m=2^{b_1}\times3^{b_2}\times...\times P^{b_n}$

where $\left \{ a_n \right \}\in \mathbb{N}\cup \left \{ 0 \right \}$ , $\left \{ b_n \right \}\in \mathbb{N}\cup \left \{ 0 \right \}$ and $P$ is a prime.

WLOG, assuming that $a_1\leq b_1, a_2\leq b_2,...,a_n\leq b_n$ ,

The $\operatorname{lcm}(l,m)$ can be obtained by choosing the highest power of each prime factor from either of the numbers. $\operatorname{lcm} (l,m)=2^{b_1}\times 3^{b_2}\times...P^{b_n}$

On the other hand, the $\gcd (l,m)$ is obtained by choosing the smallest power of each prime factor from either of the numbers.

$\gcd (l,m)=2^{a_1}\times3^{a_2}\times...\times P^{a_n}$

Notice that

$\gcd (l,m)\times \operatorname {lcm}(l,m)= 2^{a_1+b_1}\times 3^{a_2+b_2}\times ... \times P^{a_n+b_n}$

which is equivalent to

$l \times m=2^{a_1+b_1}\times 3^{a_2+b_2}\times...\times P^{a_n+b_n}$

$\therefore\gcd(l,m)\times \operatorname {lcm}(l,m)=l\times m$

I may have made some mistakes in my proof. Please do correct me if you spotted any.

Ho Wei Haw - 7 years, 9 months ago

See http://www.proofwiki.org/wiki/ProductofGCDandLCM for a much neater, albeit trickier, proof of the above.

Siddharth Prasad - 7 years, 9 months ago

You can refer to the GCD/LCM writeup in the Olympiad section.

This is a basic interesting fact which relates these two.

Is there a similar equation for the three variable case? Why, or why not?

Calvin Lin Staff - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$