AMC8 2003

AMC8 2003 · Q19

AMC8 2003 · Q19. It mainly tests GCD & LCM.

How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors?

1000 到 2000 之间有多少个整数同时被 15、20 和 25 整除？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (C)

正确答案：（C）

Solution

A number with 15, 20 and 25 as factors must be divisible by their least common multiple (LCM). Because $15 = 3 \times 5$, $20 = 2^2 \times 5$, and $25 = 5^2$, the LCM of 15, 20 and 25 is $2^2 \times 3 \times 5^2 = 300$. There are three multiples of 300 between 1000 and 2000: 1200, 1500 and 1800.

一个被 15、20 和 25 整除的数必须能被它们的最小公倍数（LCM）整除。因为 $15 = 3 \times 5$，$20 = 2^2 \times 5$，$25 = 5^2$，15、20 和 25 的 LCM 是 $2^2 \times 3 \times 5^2 = 300$。1000 到 2000 之间有三个 300 的倍数：1200、1500 和 1800。

Topics

NT.003 GCD & LCM