# Least common multiple

*Common multiple of some numbers.*

Least common multiple (LCM). Finding LCM.

Least common multiple (LCM). Finding LCM.

*
Common multiple
*
of some numbers is called a number, which is divisible by each of them. For example, numbers 9, 18 and
45 have as a common multiple 180. But 90 and 360 are also theirs common multiples. Among all common multiples there is always the least one,
in our case this is 90. This number is called a
**
least common multiple
**
(LCM).

To find a
**
least common multiple
**
(LCM) of some numbers it is necessary:

1) to express each of the numbers as a product of its
*
prime factors
*
, for example:

504 =
2
·
2
·
2
·
3
·
3
·
7 ,

2) to write
*
powers of all prime factors
*
in the factorization as:

504 =
2
·
2
·
2
·
3
·
3
·
7 = 2
^{
3
}
·
3
^{
2
}
·
7
^{
1
}
,

3) to write out
*
all prime factors
*
, presented at least in one of these numbers;

4) to take
*
the greatest power
*
of each of them, meeting in the factorizations;

5) to multiply these powers.

E x a m p l e . Find LCM for numbers: 168, 180 and 3024.

S o l u t i o n . 168 = 2
·
2
·
2
·
3
·
7 = 2
^{
3
}
·
3
^{
1
}
·
7
^{
1
}
,

180 = 2
·
2
·
3
·
3
·
5 = 2
^{
2
}
·
3
^{
2
}
·
5
^{
1
}
,

3024 = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7
= 2
^{
4
}
· 3
^{
3
}
· 7
^{
1
}
**
**
.

Write out the greatest powers of all prime factors: 2

^{ 4 }, 3

^{ 3 }, 5

^{ 1 }, 7

^{ 1 }

and multiply them:

LCM = 2

^{ 4 }· 3

^{ 3 }· 5 · 7 = 15120 .