Question

$1000$ 和 $2000$ 之间有多少个整数同时以 $15$ ， $20$ 和 $25$ 为因数？

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

$1$

$2$

$3$

$4$

$5$

使用 $15$ 、 $20$ 、 $25$ 的最小公倍数

Use the least common multiple of $15,$ $20,$ and $25.$

数出 $1000$ 和 $2000$ 之间该最小公倍数的倍数

Count the multiples of that LCM between $1000$ and $2000.$

如果一个数 $x$ 以这三个数为因数，那么它们的最小公倍数也必须整除 $x$ 。

这些数的质因数分解为 $15 = 3 \cdot 5\text{，}$ $20 = 2^2 \cdot 5\text{，}$ $25 = 5^2\text{。}$

因此最小公倍数为 $2^2 \cdot 3 \cdot 5^2 = 300\text{。}$

$300$ 的倍数中，位于 $1000$ 和 $2000$ 之间的是 $1200$ 、 $1500$ 和 $1800$ 。

所以正确答案是 C。

If a number $x$ has these three numbers as factors, then their least common multiple must also divide $x.$

These numbers have the following prime factorizations: $15 = 3 \cdot 5,$ $20 = 2^2 \cdot 5,$ $25 = 5^2.$

From these values, we get that the least common multiple is $2^2 \cdot 3 \cdot 5^2 = 300.$

Therefore, the multiples of $300$ between $1000$ and $2000$ are $1200,$ $1500,$ and $1800.$

Thus, C is the correct answer.