AMC8 1991

AMC8 1991 · Q9

AMC8 1991 · Q9. It mainly tests Inclusion–exclusion (basic), Divisibility & factors.

How many whole numbers from 1 through 46 are divisible by either 3 or 5 or both?

从 1 到 46 中，有多少个整数能被 3 或 5（或两者）整除？

(A) 18 18

(B) 21 21

(D) 25 25

(E) 27 27

Answer

Correct choice: (B)

正确答案：（B）

Solution

A number is divisible by 3 if it is a multiple of 3, and it is divisible by 5 if it is a multiple of 5. There are 15 multiples of 3, and 9 multiples of 5 which are whole numbers less than 46. However, 3 numbers (15, 30 and 45) which are divisible by both 3 and 5 have been counted twice. Thus the total number which are divisible by either 3 or 5 or both is $15 + 9 - 3 = 21$.

能被 3 整除的数是 3 的倍数，能被 5 整除的是 5 的倍数。小于 46 的 3 的倍数有 15 个，5 的倍数有 9 个。但是 3 个数（15、30 和 45）被 3 和 5 同时整除，被重复计算了。因此，总数是 $15 + 9 - 3 = 21$。

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