# What is the LCM of 10, 15, 20 and 30?

Mar 29, 2016

$60$

#### Explanation:

First, write out the prime factorization of each number

• $10 = 2 \times 5$
• $15 = 3 \times 5$
• $20 = {2}^{2} \times 5$
• $30 = 2 \times 3 \times 5$

We can rewrite the above with more clarity as

• $10 = {2}^{1} \times {3}^{0} \times {5}^{\textcolor{b l u e}{1}}$
• $15 = {2}^{0} \times {3}^{\textcolor{b l u e}{1}} \times {5}^{\textcolor{b l u e}{1}}$
• $20 = {2}^{\textcolor{b l u e}{2}} \times {3}^{0} \times {5}^{\textcolor{b l u e}{1}}$
• $30 = {2}^{1} \times {3}^{\textcolor{b l u e}{1}} \times {5}^{\textcolor{b l u e}{1}}$

For each prime factor, take the one with the highest exponent. 2 is raised to the power of 2 in 20. 3 and 5 have both a maximum exponent of 1. Refer to the $\textcolor{b l u e}{\text{blue}}$ colored exponents above.

Therefore,

$\text{LCM} = {2}^{2} \times {3}^{1} \times {5}^{1}$

$= 60$

This algorithm is guaranteed to generate the least common multiple.

Jul 24, 2017

$L C M = 60$

#### Explanation:

The first thing to notice is that we do not need to consider $10 \mathmr{and} 15$ at all because they are factors of $20 \mathmr{and} 30$ respectively.

We only need to find the LCM of $\textcolor{b l u e}{20 \mathmr{and} 30}$

You should be very familiar with these two numbers and their multiples.

The quickest and easiest method is to consider the multiples of the bigger one (30), until you find the first one which is a multiple of $20$.

The multiples of $30$ are: $30 , \textcolor{m a \ge n t a}{60} , 90 , 120 \ldots$
$\textcolor{w h i t e}{w w w w w w w w w w . w w w w w} \uparrow$
$\textcolor{w h i t e}{w w w w w w w w w w . w w w w} 20 \times 3$

$60$ is the multiple we need. It is divisible by $10 , 15 , 20 \mathmr{and} 30$

If the given numbers had been bigger or with less obvious factors and multiples, then I would have used the method of prime factors, but this one can be found mentally.