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# If a is divisible by 9!, then a/6 must be

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Math Expert
Joined: 02 Sep 2009
Posts: 45423
If a is divisible by 9!, then a/6 must be [#permalink]

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02 Dec 2016, 06:57
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5% (low)

Question Stats:

79% (00:37) correct 21% (00:37) wrong based on 95 sessions

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If a is divisible by 9!, then a/6 must be

I. divisible by 3
II. divisible by 14
III. a multiple of 100

A. None
B. II only
C. III only
D. I and II only
E. I, II, and III

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Senior CR Moderator
Status: Long way to go!
Joined: 10 Oct 2016
Posts: 1378
Location: Viet Nam
If a is divisible by 9!, then a/6 must be [#permalink]

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Updated on: 02 Dec 2016, 09:57
Bunuel wrote:
If a is divisible by 9!, then a/6 must be

I. divisible by 3
II. divisible by 14
III. a multiple of 100

A. None
B. II only
C. III only
D. I and II only
E. I, II, and III

$$\frac{a}{6}=\frac{2\times 3 \times ... \times 8 \times 9}{6}=4 \times 5 \times 6 \times 7 \times 8 \times 9$$

(I) $$\frac{a}{6}$$ has factor $$6=2 \times 3$$ so $$a/6$$ is divisible by 3. True

(II) $$\frac{a}{6}$$ has factor $$7 \times 4=2\times 14$$ is divisible by 14. True.

(III) $$100=2^2\times 5^2$$. However $$\frac{a}{6}$$ isn't divisible by $$5^2$$ so (III) is not true.

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Originally posted by broall on 02 Dec 2016, 07:31.
Last edited by broall on 02 Dec 2016, 09:57, edited 1 time in total.
Manager
Joined: 02 Nov 2013
Posts: 93
Location: India
Re: If a is divisible by 9!, then a/6 must be [#permalink]

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02 Dec 2016, 08:15
Let's say a has minimum value as 9!, then

a/6 = 9*8*7*5*4*3*2*1. Let go to answer choice now,

I. divisible by 3 = Yes. possible. As a/6 is divisible by 3.
II. divisible by 14 = Yes. Again Possible. As a/6 is divisible by 7*2.
III. a multiple of 100 = Not possible. does not have number which makes it to 100.
Math Expert
Joined: 02 Sep 2009
Posts: 45423
Re: If a is divisible by 9!, then a/6 must be [#permalink]

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03 Dec 2017, 08:22
Bunuel wrote:
If a is divisible by 9!, then a/6 must be

I. divisible by 3
II. divisible by 14
III. a multiple of 100

A. None
B. II only
C. III only
D. I and II only
E. I, II, and III

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Re: If a is divisible by 9!, then a/6 must be   [#permalink] 03 Dec 2017, 08:22
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