AMC 8 2003 Problem 19

AMC8 2003 年第 19 题

整除与因数★★★☆☆

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

1000 和 2000 之间有多少个整数，满足 15，20 和 25 都是它的因子？

正确答案 · Correct Answer

解析 · Solution

Solution 1

Find the least common multiple of 15, 20, 25 by turning the numbers into their prime factorization.

15 = 3 \cdot 5, 20 = 2^2 \cdot 5, 25 = 5^2

Gather all necessary multiples

3, 2^2, 5^2

when multiplied gets 300. The multiples of 300 - 300, 600, 900, 1200, 1500, 1800, 2100. The number of multiples between 1000 and 2000 is 3.

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