Try to prove it!

I saw some formulas regarding HCF and LCM and I am unable to prove them still.

Try if you can, if yes then please write it in the comments

$LCM(a,b,c)=\frac{a×b×c×HCF(a,b,c)}{HCF(a,b)×HCF(b,c)×HCF(c,a)}$

$HCF(a,b,c)=\frac{a×b×c×LCM(a,b,c)}{LCM(a,b)×LCM(b,c)×LCM(c,a)}$

$LCM of fractions=\frac{LCM of numerators}{HCF of denominators}$

$HCF of fractions=\frac{HCF of numerators}{LCM of denominators}$

Try to prove them

#NumberTheory

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Comments

I can prove the first two formulas. Consider the primes that divide $a$ , $b$ , and $c$ . If we show that the powers of each prime on both sides of the equation are the same, then both sides of the equation are equal. Take some prime $p$ . Say that $p^{k_{1}}|a$ , $p^{k_{2}}|b$ , and $p^{k_{3}}|c$ , with $k_{1}\leq k_{2}\leq k_{3}$ . Then, the exponent of $p$ on the left hand side is $k_{3}$ . The exponent on the right hand side is $k_{1}+k_{2}+k_{3}+k_{1}-k_{1}-k_{2}-k_{1}=k_{3}$ , so both sides are equal. The proof of the second formula is similar.

Advait Nene - 1 year, 1 month ago

Thanks for the proof

Sahar Bano - 1 year, 1 month 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}`	$\boxed{123}$