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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 66% (01:44) correct 34% (01:59) wrong based on 131 sessions

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If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 3

* A solution will be posted in two days.

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Marshall & McDonough Moderator D
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Location: India
Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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(n + 4)! > (n + 2)! > (n - 1)! --> GCD of the 3 numbers = (n - 1)!

(n - 1)! = 5!
n = 6

GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4219
Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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MathRevolution wrote:
If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 3

* A solution will be posted in two days.

Notice that:
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120

So, the smallest factorial must be 5! for the GCD to be 120
The smallest of the three factorials is (n-1)!
So, we need (n-1)! to equal 5!
This means that n = 6

If n = 6, then (n+2)!, (n-1)!, and (n+4)! become 8!, 5! and 10!

Let's CONFIRM this conclusion
8! = (1)(2)(3)(4)(5)(6)(7)(8) = (120)(6)(7)(8)
5! = (1)(2)(3)(4)(5) = 120
10! = (1)(2)(3)(4)(5)(6)(7)(8)(9)(10) = (120)(6)(7)(8)(9)(10)

As you can see, 120 is a divisor of all three factorials.
In fact, 120 is the GREATEST COMMON divisor of all three factorials.

Cheers,
Brent
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Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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1
MathRevolution wrote:
If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 3

* A solution will be posted in two days.

120 = 5 x 4 x 3 x 2 x 1

No try to substitute the options in place of n in the form

The problem states that 120 is the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! , so the three factorials must be greater than 120 ( Rather 5! )

Now find out -

A. 4

(n+2)!, (n-1)!, and (n+4)! => 6! , 3! , 8!

B. 5

(n+2)!, (n-1)!, and (n+4)! => 7! , 4! , 9!

C. 6

(n+2)!, (n-1)!, and (n+4)! => 8! ,5! , 10!

D. 7

(n+2)!, (n-1)!, and (n+4)! => 9! , 8! , 11!

E. 3

(n+2)!, (n-1)!, and (n+4)! => 5! , 2! , 7!

Now check we have 2 contenders (C) and (D)

While we are confident about (C) lets check why not (D)

9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
11! = 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Now I think you got it , yes the least common factor of 9! , 8! , 11! is 8! not 5!

Phew , lengthy post HUH...
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Thanks and Regards

Abhishek....

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
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Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 3

==>(n+2)!=(n+2)(n+1)n(n-1)!, (n-1)!,
and (n+4)!=(n+4)(n+3)(n+2)(n+1)n(n-1)!.
Then, the greatest common divisor becomes (n-1)!=120.
n-1=5 -> n=6.
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Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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Well , the case with factorials is that:

Factorial of any positive integer is Always a factor of its successive factorials (eg, 4! will be a factor of 5!, 6!, 7!, 8!,... and so on)

And Factorial of any positive integer is Always a multiple of previous factorials (eg, 6! is a multiple of 5!, 4!, 3!, ... and so on)

So, on the basis of this information we can say that if we are given various factorials, the smallest of them will be the GCD/HCF of the lot and the largest of them will be the LCM of the lot.
Eg - if we are given 7!, 62!, 35! and 19! - then 7! will be their GCD and 62! will be their LCM.

Now, we are given (n+2)!, (n-1)! and (n+4)!. Their GCD will be the smallest one, i.e., (n-1)!

So, (n-1)! = 120, which is 5!
Thus n-1 = 5 or n = 6.

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Re: If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, w  [#permalink]

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MathRevolution wrote:
If the greatest common divisor of (n+2)!, (n-1)!, and (n+4)! is 120, what is the value of n?

A. 4
B. 5
C. 6
D. 7
E. 3

We need the following fact:

If n < m, then n! is a divisor of m!.

For example, 4! is a divisor of 5! and 3! is a divisor of 8!. Now, since 5! = 120, we need n to be at least 6, in order to make (n - 1)! divisible by 5!. So this eliminates choices A, B, and E. Let’s check the remaining two choices:

If n = 6, then (n+2)! = 8!, (n-1)! = 5!, and (n+4)! = 9!. Since 5! Is the smallest of them, 5! = 120 would be the greatest common divisor.

If n = 7, then (n+2)! = 9!, (n-1)! = 6!, and (n+4)! = 10!. Since 6! Is the smallest of them, 6! = 720 would be the greatest common divisor.

Therefore, we see that n must be 6.

Alternate Solution:

Since (n - 1)! is a divisor of both (n + 2)! and (n + 4)!; GCD of (n + 4)!, (n + 2)! and (n - 1)! is (n - 1)!. We are told that the GCD of these three expressions is 120, thus (n - 1)! = 120. Since 120 = 5!, we have n - 1 = 5 and so, n = 6.

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