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YB10
Joined: 13 Feb 2023
Last visit: 03 Jan 2025
Posts: 59
Given Kudos: 30
Location: India
Schools: Darden '26 (II) Stern '27 (A)
GMAT Focus 1: 705 Q88 V86 DI85 
GPA: 9.48
23 Sep 2023, 20:44
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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47% (02:55) correct
53%
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based on 79
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If n is an integer, is n!/(n - 3)! divisible by 24?
(1) n(n + 1) is divisible by 4
(2) (n - 2)(n - 3) is divisible by 6
(1) n(n + 1) is divisible by 4
(2) (n - 2)(n - 3) is divisible by 6
23 Sep 2023, 22:38
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YB10
\(\frac{n!}{(n-3)!} = \frac{n(n-1)(n-2)(n-3)!}{(n-3)!}\)
Canceling \((n-3)!\) from numerator and denominator we get -
\(n(n-1)(n-2)\)
This represents three consecutive numbers, so \((n)(n-1)(n-2)\) will always be divisible by \(3\). We have to find if the number is divisible by \(8\) to prove that \(\frac{n!}{(n-3)!}\) is divisible by \(24\)
Effective question - Is \((n)(n-1)(n-2)\) divisible by 8 ?
Statement 1
1) n(n+1) is divisible by 4
Case 1: \(n\) is even
If \(n\) is even, \((n+1)\) is odd. Hence, for the statement 1 to hold true, \(n\) must be divisible by 4. We can conclude that \((n-2)\) is also even. Hence, \((n)(n-1)(n-2)\) is divisible by 8. For example, if n = 8, the response to the question "Is \((n)(n-1)(n-2)\) divisible by 8 ?" is Yes.
Case 2: \(n\) is odd
If \(n\) is odd, \((n+1)\) is even. Hence, for the statement 1 to hold true, \(n+1\) must be divisible by 4. While we know that the (n+1) is divisible by 4, we don't know anything about (n-1). (n-1) may or may not be divisible by 8.
For example
n = 3 ⇒ \(n(n+1)\) = 3 * 4 ; 12 is divisible by 4
\((n)(n-1)(n-2)\) = 3 * 2 * 1 ; 6 is not divisble by 24.
As we are getting two conflicting answers, we can conclude that the statement is not sufficient and we can eliminate A and D.
Statement 2
2) (n-2)(n-3) is divisible by 6
We can take some clues from statement 1, and first check with \(n = 3\).
If n = 3, (n-2)(n-3) = 0 ⇒ 0 is divisible by 6.
We know that if n = 3, the response to the question "Is \((n)(n-1)(n-2)\) divisible by 8 ?" is No.
If (n-2) is even, and let's assume (n-2) = 6 → n = 8. If n = 8, the response to the question "Is \((n)(n-1)(n-2)\) divisible by 8 ?" is Yes.
As we are getting two conflicting answers, we can eliminate B.
Combined
The statements combined don't help either as n = 3 and n = 8 satisfy both the statements, however, we have a conflicting response to that target question using the values.
n = 3 ⇒ Is \((n)(n-1)(n-2)\) divisible by 8 ? -- No
n = 8 ⇒ Is \((n)(n-1)(n-2)\) divisible by 8 ? -- Yes
Option E
14 Apr 2025, 08:58
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