M21-29

Question

How many integers between 18 and 70, inclusive, are divisible by 7 or 9, but not by both ?

A. 11
B. 12
C. 13
D. 14
E. 15

Bunuel · Accepted Answer

Official Solution:

How many integers between 18 and 70, inclusive, are divisible by 7 or 9, but not by both ?

A. 11
B. 12
C. 13
D. 14
E. 15

The number of multiples of an integer in a range is given by the formula  \frac{	ext{last multiple in the range - first multiple in the range}}{	ext{multiple}}+1 . Hence:
 
• Number of multiples of 7 in the given range  \frac{(	ext{last-first})}{	ext{multiple}}+1=\frac{(70-21)}{7}+1=8 ;
 
• Number of multiples of 9 in the given range  \frac{(	ext{last-first})}{	ext{multiple}}+1=\frac{(63-18)}{9}+1=6 ;
 
• Number of multiples of both 7 and 9 is 1, which is  7*9=63 . 
 
Note that the number 63 is a multiple of both 7 and 9 and is included in the count of multiples of 7 and the count of multiples of 9. Since we don't want it to be counted at all, we need to subtract it once from the total count of multiples of 7, and once from the total count of multiples of 9.
 
Therefore, the number of positive integers that are either multiples of 7 or multiples of 9, but not multiples of both, in the given range is  (8-1)+(6-1)=12 .

Answer: B

Nevernevergiveup · Answer

Between 18 and 70 inclusive

multiples of 7 consist {21,28,35,42,49,56,63,70}

multiples of 9 {18,27,36,45,54,63}

since we need numbers that are divisible by either 7 or 9 but not both.

9*7=63  is to be removed from both the sets.

final set {18,21,27,28,35,36,42,45,49,54,56,70}

no of terms in final set = 12

benejo · Answer

Good question. I subtracted only 1 from 14. Dumb mistake :/

dcummins · Answer

Made the common silly mistake of only deducting 63 from 1 set.

ChuHoaiNam · Answer

This is a high quality question. Let image in the test day, the question is: " How many integers between 50 and 1000, inclusive are divisible by either 7 or 9 but not both"
Can you list down all numbers in 2 mins? Of course not, the method of Bunuel is the best way to solve.

coreyander · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

comptee · Answer

I think this is a  high-quality  question and I agree with explanation.

blonpina · Answer

I think this is a  high-quality  question and I agree with explanation.

phmahi1997 · Answer

Damn! I made such a silly mistake. Interestingly, some people, like me, made the same mistake.

Here's what went wrong:

I calculated for 7 and 9, then for both 7 & 9, which is 63.

But I deducted only one time from the solution. But in reality, 63 counted for two times, once when we counted 7 and the other time in 9.

Mohak01 · Answer

Quality problem. This questions looks more like OG than OG questions itself.

Bunuel · Answer

Thank you! We try to test relevant GMAT topics in GMAT Club Tests and also include twists that help develop the critical skills needed for the real exam.

M21-29

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