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Number Theory

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Moderate

Mersenne Number GCD and LCM

NUMTH-SBJWNS

Mersenne number is a positive integer of the form $M_n:=2^n-1$ for a positive integer $n$.
We write $(m,n)$ for the greatest common divisor of $m$ and $n$, and $[m,n]$ for the least common multiple of $m$ and $n$.

What is the least common multiple of $M_m$ and $M_n$? Select ONE answer which is always true.

A

$M_{mn}$

B

$M_{[m,n]}$

C

$\frac{M_m M_n}{M_{(m,n)}}$

D

$\begin{cases} M_m M_n &\mbox{ if $m$ and $n$ are distinct, }\\\ M_m &\mbox{ otherwise.}\end{cases}$

E

$\begin{cases} M_{[m,n]} &\mbox{ if $m$ and $n$ are distinct, }\\\ M_m &\mbox{ otherwise.}\end{cases}$