Least common multiple
Common multiple of some numbers. 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 .
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