Problem Solving

how many such numbers are there between 1 and 500 which are multiples of 3 and 7 or of both 3 and 7???

211
212
213
214

AIM TO CRACK GMAT wrote:how many such numbers are there between 1 and 500 which are multiples of 3 and 7 or of both 3 and 7???

211
212
213
214

Multiples of 3:
3,6,9,....,498
Total = (498 - 3)/3 + 1 = 495/3 + 1 = 165+1 = 166

Multiples of 7:
7,14,21,...,497
Total = (497 - 7)/7 + 1 = 490/7 + 1 = 70 + 1 = 71

Multiples of (3*7) = 21:
21,42,...,483
Total = (483 - 21)/21 + 1 = 462/21 + 1 = 22 + 1 = 23

Grand total = 166 + 71 - 23 = 214

Choose D

Cheers

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.

I think this question would be better worded as:

how many such numbers are there between 1 and 500 which are multiples of 3 OR 7, or of both 3 and 7???

The distinction between AND and OR makes a real difference in meaning.

Otherwise, the answer is 23.