If (5^4 - 1)/n is an integer an n is an integer, then n could be each

Question

If  \frac{(5^4 - 1)}{n}  is an integer an n is an integer, then n could be each of the following EXCEPT

(A) 4
(B) 6
(C) 13
(D) 25
(E) 26

pushpitkc · Answer

5^4 - 1  can be simplied as  625 - 1 = 624

Of the available answer options, Only  Option D(25)  will not divide 624 and is our answer.

PS 25 =  5^2  and 5 divides a number ending in 0 or 5. The traditional method of solving this problem is 
to prime-factorize 624 and eliminate all those options which can be formed from the prime-factors.

AsyaWander · Answer

My answer choice is D

The statement can be re-written as follows:
1. (5^2-1)(5^2+1)/n
2. 24*26/n
3. From the given choices 4 and 6 are factors of 24, 13 and 26 are factors of 26

chetan2u · Answer

you can simplify the equation
 \frac{(5^4 - 1)}{n}=\frac{(5^2-1)(5^2+1)}{n}=\frac{(25-1)(25+1)}{n}=\frac{24*26}{n} 
so n has to be a FACTOR of 24*26...

clearly in choices 25 has a 5 in it hence not a factor of 24*26
 
ALSO without solving we can see 5^4-1 cannot be a multiple of 4, so n cannot be a multiple of 5 hence 25 is the answer

D

JeffTargetTestPrep · Answer

First, we recognize that the numerator, 5^4 - 1, is a difference of squares, and we factor it as (5^2 - 1)(5^2 + 1). Then we evaluate the result:

5^4 - 1 = (5^2 - 1)(5^2 + 1) = 24 x 26 = 4 x 6 x 2 x 13.

Since the fraction (5^4 - 1)/n is an integer, we see that n can be any of the answer choices except 25.

Answer: D

TaN1213 · Answer

Hello Chetan,

I did not get the highlighted part. 5^4-1 must be a multiple of 4 in my opinion. Please let me know where I am going wrong. Thank you.

akadiyan · Answer

If (5^4-1)/n is a integer, then n could be each of the value except

5^4  = 625 , 625-1 = 624

Out of answer choice, only 25 does not result in integer.

Ans:   D

EgmatQuantExpert · Answer

Just to add with the existing discussion:

Following the options, we do not need to factorize the expression. Use of unit digit will be sufficient to get the answer.

•  5^n  will always end with the unit digit 5
• Therefore,  5^n – 1  will always end with the unit digit 5 – 1 = 4

Now, a number ending with the digit 4 can never be divisible by 25

Hence, the correct answer choice is Option D.

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If (5^4 - 1)/n is an integer an n is an integer, then n could be each

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If (5^4 - 1)/n is an integer an n is an integer, then n could be each

Prep Toolkit

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