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

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

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New post 14 Dec 2016, 05:53
1
3
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

72% (01:14) correct 28% (00:58) wrong based on 154 sessions

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

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New post Updated on: 14 Dec 2016, 08:33
1
\(\frac{a}{5!} = integer\)

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"

Hence answer C

Originally posted by vitaliyGMAT on 14 Dec 2016, 06:25.
Last edited by vitaliyGMAT on 14 Dec 2016, 08:33, 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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New post 14 Dec 2016, 08:16
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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New post 14 Dec 2016, 12:00
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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New post 03 Dec 2017, 08:21
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]

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New post 19 Dec 2018, 19:31
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Re: If a is divisible by 5!, then a/2 must be   [#permalink] 19 Dec 2018, 19:31
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