# The least common multiple of 84 and N is 504. How to find " N " ?

Jun 4, 2016

$N = 72$ or $N = 504$

#### Explanation:

The least common multiple (LCM) of two integers $a$ and $b$ is the least number $c$ such that $a n = c$ and $b m = c$ for some integers $n$ and $m$.

We can find the LCM of two integers by looking at their prime factorizations, and then taking the product of the least number of primes needed to "contain" both. For example, to find the least common multiple of $28$ and $30$, we note that

$28 = {2}^{2} \cdot 7$
and
$30 = 2 \cdot 3 \cdot 5$

In order to be divisible by $28$, the LCM must have ${2}^{2}$ as a factor. This also takes care of the $2$ in $30$. In order to be divisible by $30$, it must also have $5$ as factor. Finally, it must have $7$ as a factor, too, to be divisible by $28$. Thus, the LCM of $28$ and $30$ is

${2}^{2} \cdot 5 \cdot 7 \cdot 3 = 420$

If we look at the prime factorizations of $84$ and $504$, we have

$84 = {2}^{2} \cdot 3 \cdot 7$
and
$504 = {2}^{3} \cdot {3}^{2} \cdot 7$

Working backwards, we know that ${2}^{3}$ must be a factor of $N$, or else the LCM would only need ${2}^{2}$ as a factor. Similarly, we know ${3}^{2}$ is a factor of $N$ or else the LCM would only need $3$ as a factor. Then, as $7$, the only other factor of the LCM, is needed for $84$, $N$ may or may not have $7$ as a factor. Thus, the two possibilities for $N$ are:

$N = {2}^{3} \cdot {3}^{2} = 72$
or
$N = {2}^{3} \cdot {3}^{2} \cdot 7 = 504$