# Find the smallest integer for which 756k is a multiple of 1584?

Nov 28, 2016

The smallest integer $k$ could be is $44$ so that $756 k$ is a multiple of $1584$.

#### Explanation:

The factors of $756$ are

$756 = 2 \times 2 \times 3 \times 3 \times 3 \times 7$ and

factors of $1584$ are

$1584 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 11$

As $756 k$ is a multiple of $1584$, it must have at least - two additional $2 s$ and one additional $11$.

Hence, $k$ should be minimum $2 \times 2 \times 11 = 44$ so that $756 k$ is a multiple of $1584$.