AIM TO CRACK GMAT wrote:How many integers between 1 and 500 are multiples of 3 or 7 or of both 3 and 7?
211
212
213
214
SUBTRACT THE OVERLAP.
Total = Multiples of 3 + Multiples of 7 - Multiples of 21.
When we count the multiples of 3 and the multiples of 7, the OVERLAP between the two groups -- the multiples of 21 -- are counted TWICE.
So that the multiples of 21 are not double-counted, we must subtract them from the total.
The endpoints of the range are 1 and 500, implying a total of 500 integers.
Since neither endpoint is divisible by 3 or 7, we can can quickly count the multiples of 3, 7 and 21 by determining how many times 3, 7 and 21 each divide into 500.
Multiples of 3 between 1 and 500:
500/3 = 166.
Multiples of 7 between 1 and 500:
500/7 = 71.
Multiples of 21 between 1 and 500:
500/21 = 23.
Thus:
Total = 166 + 71 - 23 = 214.
The correct answer is
D.
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