# What is the least common multiple (LCM) of 8, 10 and 12?

Jan 17, 2017

The least common multiple of 10, 12, and 8 is 120.

#### Explanation:

You know that twelve and ten have a common multiple every sixty. And you have to know that every forty you can have a common multiple of ten and eight. So the least common multiple on forty and sixty is one hundred and twenty.

Jan 17, 2017

$120$

#### Explanation:

Write each number in prime factor form

$10 = 2 \cdot 5$

$12 = 2 \cdot 2 \cdot 3$

$8 = 2 \cdot 2 \cdot 2$

The LCM is

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 120$

Jan 18, 2017

LCM is 120

#### Explanation:

$\textcolor{b l u e}{\text{Method 1}}$

Just looking at and considering the numbers:

$\textcolor{b r o w n}{\text{Observation 1}}$

A multiple of 10 by any whole positive number will have 0 as the last digit. So the least common multiple has 0 for the last digit.
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$\textcolor{b r o w n}{\text{Observation 2}}$
All the numbers are even so the common multiple has to be even thus divisible by 2. This is what makes the derivation more straightforward.
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$\textcolor{b r o w n}{\text{Observation 3}}$

Consider the 2 from 12: the multiple of 12 has to end in 0

note the least value times 12 to give 0 as a last digit is 5 and $5 \times 12 = 60$

But 60 does not divide exactly by 8 so using observation 2 multiply 60 by 2 giving 120
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$\textcolor{b l u e}{\text{Method 2}}$

You would probably find it very useful to remember some of the prime numbers. Gradually build your memory up to the number 101.
Do an image search in google for Prime Factors.

The first few are: 2;3;5;7;11;13
The method with prime factors is to use the maximum count of factors for any one the numbers given and multiply them all together. So we have:

So we have:$\text{ } \left(2 \times 2 \times 2\right) \times \left(3\right) \times \left(5\right) = 120$