初等代数学:
最小公倍数

1.  
4,64,6
  解答
2.  
8,128,12
  解答
3.  
14,7,2814,7,28
  解答
4.  
3,4,83,4,8
  解答
5.  
2,6,282,6,28
  解答
6.  
13,213,2
  解答

Lowest Common Multiple - Introduction

A multiple of a number is the result of multipling that number by an integer (whole number). Consider the multiples of
33
and
44
:

Multiples of
33

3×1=33\times 1=3

3×2=63\times 2=6

3×3=93\times 3=9

3×4=123\times 4=12

3×5=153\times 5=15

3×6=183\times 6=18

3×7=213\times 7=21

3×8=243\times 8=24

Multiples of
44

4×1=44\times 1=4

4×2=84\times 2=8

4×3=124\times 3=12

4×4=164\times 4=16

4×5=204\times 5=20

4×6=244\times 6=24

4×7=284\times 7=28

4×8=324\times 8=32

Notice that there are certain multiples that are the same for both
33
and
44
, since both can multiply to
1212
and
2424
. For many math problems, it’s helpful to know the smallest multiple that a set of numbers share. This is called the least common multiple, or lowest common multiple, also written as LCM. In this case, the least common multiple of
33
and
44
is
1212
.

Methods of Finding the Lowest Common Multiple

There are two methods to help you easily find the LCM.

Method 1: Listing Multiples

This is the method we used above. Take two (or more) numbers and list their multiple lists until you find a common answer. Consider the lists of
44
,
77
and
88
.

Multiples of
44

4×1=44\times 1=4

4×2=84\times 2=8

4×3=124\times 3=12

4×4=164\times 4=16

4×5=204\times 5=20

4×6=244\times 6=24

4×7=284\times 7=28

4×8=324\times 8=32

4×9=364\times 9=36

4×10=404\times 10=40

4×11=444\times 11=44

4×12=484\times 12=48

4×13=524\times 13=52

Multiples of
77

7×1=77\times 1=7

7×2=147\times 2=14

7×3=217\times 3=21

7×4=287\times 4=28

7×5=357\times 5=35

7×6=427\times 6=42

7×7=497\times 7=49

7×8=567\times 8=56

7×9=637\times 9=63

7×10=707\times 10=70

7×11=777\times 11=77

7×12=847\times 12=84

Multiples of
88

8×1=88\times 1=8

8×2=168\times 2=16

8×3=248\times 3=24

8×4=328\times 4=32

8×5=408\times 5=40

8×6=488\times 6=48

8×7=568\times 7=56

8×8=648\times 8=64

8×9=728\times 9=72

8×10=808\times 10=80

8×11=888\times 11=88

8×12=968\times 12=96

While
44
and
88
share multiples like
1616
,
2424
,
3232
and
2020
, the lowest common multiple for all three numbers is
5656
.

Method 2: Prime Factors

Start by listing the prime factors of each number. Prime factors are the set of prime numbers that, when multiplied, give the original number. Let’s use this method to find the LCM of
2020
and
2525
.
Prime factors of
2020
—
5×2×25\times 2\times 2
Prime factors of
2525
—
5×55\times 5
Next, multiply each factor the most number of times it appears in either prime factorization. This means we need to multiply
2×2×5×52\times 2\times 5\times 5
because
22
occurs twice in one set of prime factors and
55
occurs twice in the other. The remaining
55
in the first set of factors does not need to be multiplied. This gives us
100100
, which is the lowest common multiple for both
2020
and
2525
.

What's Next

Ready to give it a try? Use either method to calculate the LCM in our practice problems at the top of this page.
Our LCM problems are designed to help students quickly grasp and master the skill of discovering least common multiples for any set of numbers. You can also acquire skills in other topics via our practice problems. We hope that through these problems and solutions, you will be empowered with the math concepts that you need to succeed.
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