# What is the LCM of 10 and 14?

70

#### Explanation:

LCM, or Lowest Common Multiple, is the number which is both a multiple of 10 and 14 and is the first number that is.

Many students will look at the two numbers, in this case 10 and 14 and say: "If I multiply 10 and 14, I'll get 140 and that is for sure a multiple of 10 and 14 and so I'll just use that."

And it is true that 140 is a multiple of 10 and 14. But it isn't the lowest. So let's find the lowest.

When I do these, I like to break the numbers in question down to their prime factors:

For 10, the prime factors are 5 and 2 - our LCM will need a 5 and a 2
For 14, the prime factors are 7 and 2 - our LCM will need a 7 and a 2

So we'll need a 2, a 5, and a 7 to make our LCM:

$2 \times 5 \times 7 = 70$

And we can check this by saying:

$10 \times 7 = 70$
$14 \times 5 = 70$

The first attempt we made, with 140, wasn't the lowest because it had an extra 2 that wasn't needed:

$2 \times 5 \times 2 \times 7 = 140$

Sep 29, 2016

70

#### Explanation:

Consider the prime factors of 10 and 14.

$10 = 2 \times 5$ and $14 = 2 \times 7$

Hence the factors of the lowest common multiple must be $2 \times 5 \times 7$ - Since 2 appears in both it must occur only once in the LCM.

$\therefore L C M \left(10 , 14\right) = 2 \times 5 \times 7 = 70$

Sep 29, 2016

$70$

#### Explanation:

multiples of $10 : 10 , 20 , 30 , 40 , 50 , 60 , \textcolor{red}{70} , 80 \ldots$
multiples of $14 : 14 , 28 , 42 , 56 , \textcolor{red}{70} , 84 \ldots \ldots$

Hence, LCM of $10 \mathmr{and} 14 = 70$