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# If a is divisible by 5!, then a/2 must be

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Math Expert
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If a is divisible by 5!, then a/2 must be [#permalink]

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14 Dec 2016, 04:53
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Question Stats:

68% (00:59) correct 32% (00:36) wrong based on 77 sessions

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If a is divisible by 5!, then a/2 must be

I. a factor of 10
II. a multiple of 8
III. a multiple of 10

A. None
B. I only
C. III only
D. I and III only
E. I, II, and III
[Reveal] Spoiler: OA

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If a is divisible by 5!, then a/2 must be [#permalink]

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14 Dec 2016, 05:25
$$\frac{a}{5!} = integer$$

in other words $$a$$ is a multiple of $$5!$$

$$a = 120x = 2^3*3*5*x$$

$$\frac{a}{2} = 2^2*3*5*x$$

I In order to be a factor of 10 $$a$$ must divide 10 evenly - Impossible ---> $$\frac{2*5}{2^2*3*5} ≠ integer$$

II In order to be a multiple of 8 $$a$$ must have 3 factors of 2, but a is only a multiple of 4 Negative.

III $$a$$ is definitely a multiple of 10 because it has both 2 and 5 as factors.

We don't know value of x, but question asks "what must be true"

Last edited by vitaliyGMAT on 14 Dec 2016, 07:33, edited 2 times in total.

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Re: If a is divisible by 5!, then a/2 must be [#permalink]

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14 Dec 2016, 07:16
Bunuel wrote:
If a is divisible by 5!, then a/2 must be

I. a factor of 10
II. a multiple of 8
III. a multiple of 10

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

a = k*5*4*3*2

So, a/2 = k*5*4*3

I. a is a not a factor of 10, but a multiple of 10

II. a is not a multiple of 8

III. a is a multiple of 10

Hence, correct answer must be (C) III only
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Re: If a is divisible by 5!, then a/2 must be [#permalink]

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14 Dec 2016, 11:00
Option C

$$a = 5! * X = 120 * X = 2^3 * 3^1 * 5^1 * X$$

$$a/2 = 2^2 * 3^1 * 5^1 * X$$

I: $$\frac{a}{2}$$ - is not a divisor but a multiple of 10 - so NO

II: $$\frac{a}{2}$$ - can be multiple of 8 or can't be, depends on value of X - so NO

III: $$\frac{a}{2}$$ - can be written as $$2^1 * 3^1 * 10^1 * X$$ - is definitely a multiple of 10 - so YES
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Re: If a is divisible by 5!, then a/2 must be [#permalink]

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03 Dec 2017, 07:21
Bunuel wrote:
If a is divisible by 5!, then a/2 must be

I. a factor of 10
II. a multiple of 8
III. a multiple of 10

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

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Re: If a is divisible by 5!, then a/2 must be   [#permalink] 03 Dec 2017, 07:21
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