# How do you make a list of possible pairs of numbers that have a LCM of 48?

Dec 5, 2016

$14$ such pairs, which are $\left(1 , 48\right) , \left(2 , 48\right) , \left(3 , 48\right) , \left(4 , 48\right) , \left(6 , 48\right)$

$\left(8 , 48\right) , \left(12 , 48\right) , \left(16 , 48\right) , \left(24 , 48\right\} , \left(48 , 48\right) , \left(3 , 16\right) , \left(6 , 16\right) , \left(12 , 16\right) , \left(24 , 16\right)$

#### Explanation:

First of all pairing $48$ with all of its factors i.e. $\left\{1 , 2 , 3 , 4 , 6 , 8 , 12 , 16 , 24 , 48\right\}$. This gives us $10$ such pairs.

We can also add $\left(3 , 16\right)$, as they are relatively coprime. Further, selecting multiples of $3$ will also lead to the pairs $\left(6 , 16\right) , \left(12 , 16\right) , \left(24 , 16\right)$

Hence in all $14$ such pairs, which are

$\left(1 , 48\right) , \left(2 , 48\right) , \left(3 , 48\right) , \left(4 , 48\right) , \left(6 , 48\right) , \left(8 , 48\right) , \left(12 , 48\right) , \left(16 , 48\right) , \left(24 , 48\right\} , \left(48 , 48\right) , \left(3 , 16\right) , \left(6 , 16\right) , \left(12 , 16\right) , \left(24 , 16\right)$