What is the value of positive integer n ? (1) n^5! = 1 (2) n^5 = n!

From the question stem, n must be positive.

Evaluate statement (1) alone:
n = 1 or -1
From the question stem, we know it is positive, so it must be 1.
Does statement (1) alone give us enough information to find the value of n? Yes. AD.

Evaluate statement (2) alone:
n = 1
Does statement (2) alone give us enough information to find the value of n? Yes. D.

Answer choice D.

Question: n = ?

Statement 1: \(n^{5!} = 1\)

It's a typical case where \(n^{even} = 1\) which gives us several possibilities
1) n is either +1 or -1
2) exponent of n is zero

Since, exponent is non zero and n is positive therefore we are left with only one possibility of n i.e. +1

SUFFICIENT

Statement 2:\(n^5 = n!\)

We evaluate this statement as well with similar possible results for n and realize that n can only be 1 for this case hence
SUFFICIENT

Answer: Option D

Ues, I didn't read the stem well

Posted from my mobile device

Question:

What is the value of positive integer n ?
(1) \(n^5!\)=1
(2)\( n^5\)=n!

Hi all !

This question tests on several different key elements-
1. Your ability to analyze the statements
2.Application of fundamental knowledge
3. Staying in control and NOT losing the grip of the stem. Some crucial inputs in the question stem are essential to hold on to this.

This is a value based DS question and we need sufficiency or insufficiency of the statements to reach to a definite value for n.

St(1) \(n^5!\)=1
Here we have n^120 =1 or n^(even value) = 1
As an integer, n can only be a +1 or a -1
If n = -1,then the question stem is violated (n has to be positive).
Hence n is +1 and we have a definite value of n using statement 1.
Sufficient.
Eliminate B,C,E.

St(2) \(n^5\) =n!
A factorial is defined for positive integers and hence if you think of any positive integer like 2,3,4,5,..., you would notice that the equality occurs only when n =1 .
Hence n =1 (A unique answer is obtained)
Eliminate A

(option d)

Devmitra Sen
GMAT Mentor

Step 1: Analyse Question Stem

We have to find the value of n which is a positive integer.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: \(n^{5!}\)=1

Since 5! = 120, this equation can be rewritten as, \(n^{120}\) = 1.

Since the power is an even integer, the value of n could be 1 or -1. However, since the question clearly identifies n as a positive integer, n MUST be 1.

The data in statement 1 is sufficient to find a unique value of n.
Statement 1 alone is sufficient. Answer options B, C and E can be eliminated.


Statement 2: \(n^5\)=n!

Since factorial is defined for a positive integer, n has to be positive.
Additionally, the only value of n that satisfies the equation given is n = 1.

The data in statement 2 is sufficient to find a unique value of n.
Statement 2 alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

Official Solution:


What is the value of positive integer \(n\) ?

(1) \(n^{5!} = 1\)

The given equation is \(n^{120} = 1\). This implies that \(n=1\) or \(n=-1\). However, since \(n\) is specified as a positive integer, the only possibility is \(n=1\). Sufficient.

(2) \(n^5 = n!\)

It's clear that \(n=1\) satisfies the given equation. The real question now is, is it the only solution? First of all, note that if \(n\) is greater than 1, \(n!\) becomes even, therefore, if there is another solution to \(n^5 = n!\), then \(n\) must be even for \(n^5\) to also be even. Let's test some even values:

If \(n=2\), \(n^5 = 2^5 = 32\), while \(n! = 2\). The values do not match: \(n^5 > n!\).

If \(n=4\), \(n^5 = 4^5 = 2^{10} = 1024\), while \(4! = 24\). The values do not match: \(n^5 > n!\).

If \(n=6\), \(n^5 = 6^5 = 6*6*6*6*6\), while \(6! = 2*3*4*5*6\). The values do not match: \(n^5 > n!\).

If \(n=8\), \(n^5 = 8^5 = 8*8*8*8*8\), while \(8! = 2*3*4*5*6*7*8\). The values do not match. Notice that in this case, it becomes \(n^5 < n!\) instead.

For \(n\geq 8\), observe that the factorial function \(n! = 1*2*3*...*n\) grows much faster than the exponential function \(n^5 = n*n*n*n*n\). This is because when \(n \geq 8\), there are additional factors greater than or equal to \(n\) in the factorial, and their number is more than 5. These factors cause the factorial to grow much more rapidly compared to the exponential function. For example, if \(n=10\), \(n^5 = 10*10*10*10*10\), while, \(n! = 1*2*3*4*5*6*7*8*9*10=(2*5)*(3*4)*(6*7)*(8*9)*10\). Consequently, we can conclude that there is no solution for \(n \geq 8\).

Based on the above analysis, the only positive integer solution for the given equation is \(n=1\). Therefore, this statement is also sufficient.


Answer: D