Math Olympiads: Whole Numbers
Least Common Multiple
About Least Common Multiples (LCM)
The least common multiple (LCM) of two or more numbers is the smallest
integer that is a multiple of all numbers in the series.
For example, the LCM of 3, 4, and 6 is 12, since 3, 4, and 6 are all factors of
twelve. We can see this by examining the progression of multiples of each
number:
×1
×2
×3
×4
×5
×6
×7
×8
3
6
9
12
15
18
21
24
4
8
12
16
20
24
28
32
6
12
18
24
30
36
42
48
The chart shows that another common multiple of 3, 4, and 6 is 24, since 3, 4,
and 6 are all factors of 24. However, 12 is still the LCM. The LCM is necessary for
combining fractions of differing denominators—it allows us to compare apples
to apples, so to speak. Keep in mind that the LCM for two prime numbers will
always be the product of those numbers.
Another way to find the LCM is a bit more hit-or-miss. Multiply the factors
together, and then try to reduce their product until you find the LCM of your
original numbers:
3 × 4 × 6 = 72
72 reduces to 36 (multiple of 3, 4, and 6); 36 reduces to 24 (multiple of 3, 4, and 6);
24 reduces to 12 (multiple of 3, 4, and 6)...
12 reduces to 6, but 6 is only a multiple of 3 and 6...
Therefore, 12 is the LCM of 3, 4, and 6
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