2016 AMC 8 第 20 题

Question

a

和

b

的最小公倍数是

12

，

b

和

c

的最小公倍数是

15

。

a

和

c

的最小公倍数的最小可能值是多少？

The least common multiple of

a

and

b

is

12,

and the least common multiple of

b

and

c

is

15.

What is the least possible value of the least common multiple of

a

and

c?

20

30

60

120

180

答案：A

知识点：最小公倍数

难度评级：1390

小提示：

因为

b

出现在两个最小公倍数中，先考虑

b

的可能值

Since

b

appears in both least common multiples, start with possible values of

b

.

大提示：

尽量让

b

吸收

12

和

15

中共有的因子

3

。

Try to make

b

absorb the common factor

3

from

12

and

15

.

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文字解答：

b

必须同时整除

12

和

15

，所以它只能是

1

或

3

。

如果

b = 1

，则可取

a = 12

、

c = 15

，它们的最小公倍数为

60

。如果

b = 3

，则可取

a = 4

、

c = 5

，此时最小公倍数为

20

。

所以正确答案是 A。

We know that

b

has to divide both

12

and

15,

so it must equal either

1

or

3.

If

b = 1,

then

a = 12

and

c = 15,

making their least common multiple

60.

If

b = 3,

then we may take

a = 4

and

c = 5.

The least common multiple in this scenario is

20.

Thus, A is the correct answer.