# What is the least common multiple of 3,4, and 6?

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Tarik Share
Apr 10, 2016

12

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Nimo N. Share
Jan 11, 2018

12

#### Explanation:

Find the least common multiple (LCM) of 3, 4, and 6.

The LCM of two or more numbers is a number whose prime-factored form contains only the prime factors of each of the numbers.

There are two methods usually taught in school, but there are several other methods that are used, some of which are faster or easier to use, but may not be as easy to understand.

Method 1:

Factor each number into primes. One (1) is not a prime number, so it is not necessary to list it.

$4 = 2 \cdot 2.$
$6 = 3 \cdot 2.$
$L C M \left(3 , 4 , 6\right) = 3 \cdot 2 \cdot 2 = 12.$

The factors of each number can be found in in the factored LCM.

Method 2.

Make a list of a few multiples of each of the numbers. Make the lists long enough that you can find one of the numbers in each of the lists. If the list is long, there may be several such numbers, but the idea is to choose the smallest one that appears in each list.

Multiples of 3: $3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , \ldots$
Multiples of 4: $4 , 8 , 12 , 16 , 20 , 24 , \ldots$
Multiples of 6: $6 , 12 , 18 , 24 , \ldots$

Now, find the smallest ( Least ) multiple that is Common in each list of Multiples,

It is easy to pick-out the number 12 from the lists as the smallest, even though there is another number, 24, which is common to each list.

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Mar 28, 2018

The least common multiple of $3 , 4 ,$ and $6$ is $12$

#### Explanation:

The fastest way to find the least common multiple (and also the least common denominator) is usually to start naming the multiples of the largest number.

For each multiple, check to see if the other numbers can also go evenly into that multiple.

Example
Find the least common multiple of $5$ and $6$

1) Use $6$ because it's bigger.
2) Start naming the multiples of $6$

$6 \times 1 = 6$ $\leftarrow$ divisible by $6$ but not by $5$

$6 \times 2 = 12$ $\leftarrow$ divisible by $6$ but not by $5$

$6 \times 3 = 18$ $\leftarrow$ divisible by $6$ but not by $5$

$6 \times 4 = 24$ $\leftarrow$ divisible by $6$ but not by $5$

$6 \times 5 = 30$ $\leftarrow$ $\text{BINGO!}$ $30$ is evenly divisible by both $5$ and $6$

So to find the least common multiple of the numbers in the problem, you do the same procedure.

1) Use $6$ because it's the biggest
2) Start listing the multiples of $6$ until you find one that is also a multiple of $3$ and $4$

$6 \times 1 = 6$$\leftarrow$ Divisible by $3$ and by $6$ but not by $4$

$6 \times 2 = 12$ $\leftarrow$ $\text{BINGO!}$ Divisible by $3 , 4 ,$ and $6$

The LCM (and LCD) of $3 , 4$ and $6$ is 12

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