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# M21-29

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Math Expert
Joined: 02 Sep 2009
Posts: 50572

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16 Sep 2014, 00:15
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Difficulty:

65% (hard)

Question Stats:

52% (00:58) correct 48% (00:53) wrong based on 139 sessions

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

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

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Math Expert
Joined: 02 Sep 2009
Posts: 50572

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16 Sep 2014, 00:15
Official Solution:

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

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

# of multiples of 7 in the given range is $$\frac{(last-first)}{multiple}+1=\frac{(70-21)}{7}+1=8$$;

# of multiples of 9 in the given range is $$\frac{(last-first)}{multiple}+1=\frac{(63-18)}{9}+1=6$$;

# of multiples of both 7 and 9 is 1: $$7*9=63$$. Notice that 63 is counted both in 8 and 6;

So, # of multiples of either 7 or 9 but not both in the given range is $$(8-1)+(6-1)=12$$.

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Joined: 06 Jul 2015
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11 Oct 2015, 00:53
Bunuel wrote:
Official Solution:

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

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

# of multiples of 7 in the given range is $$\frac{(last-first)}{multiple}+1=\frac{(70-21)}{7}+1=8$$;

# of multiples of 9 in the given range is $$\frac{(last-first)}{multiple}+1=\frac{(63-18)}{9}+1=6$$;

# of multiples of both 7 and 9 is 1: $$7*9=63$$. Notice that 63 is counted both in 8 and 6;

So, # of multiples of either 7 or 9 but not both in the given range is $$(8-1)+(6-1)=12$$.

Hi,
Thanks for the post and knowing the fomular is very helpful.
But, just wondering why we need to add "1" after (last-first)/multiple?
# of multiples of 7 in the given range is $$\frac{(last-first)}{multiple}+1=\frac{(70-21)}{7}+1=8$$;

Thanks!
Regarda
Andy
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Joined: 18 Sep 2014
Posts: 1128
Location: India

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12 Oct 2015, 15:19
1
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
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Joined: 08 Jun 2015
Posts: 435
Location: India
GMAT 1: 640 Q48 V29
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12 Dec 2017, 02:20
The answer is B. First find the number of multiples of 7 from 18 to 70. Then the number of multiples of 9 from 18 to 63. We need to exclude the common multiple from both. 63 has been counted two times The answer is 8+6-2 . i.e. 12. Hence B.
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09 May 2018, 02:01
Good question. I subtracted only 1 from 14. Dumb mistake :/
M21-29 &nbs [#permalink] 09 May 2018, 02:01
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# M21-29

Moderators: chetan2u, Bunuel

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