Source: Official Guide Revised GRE 1st Ed. Part 8; Section 6; #21


What is the least positive integer that

What is the least positive integer that is not a factor of 25! and is not a prime number? 26, 28, 36, 56, 58

4 Explanations


ajith kumar

we can simply factorize every answer choices
26= 2*13
28= 2*2*7

And 58 is the only choice that is not a factor of 25 and is not a prime no.

May 25, 2017 • Comment

Cydney Seigerman, Magoosh Tutor

Excellent question, Ajith!

Yes, writing out the prime factorization of each answer choice is another way to approach this problem. When we do this, we see that the prime factorization of 58 contains 29, which is not a factor of 25! On the other hand, the prime factorizations of the other answer choices are all composed of factors of 25! Since 29 is not a factor of 25!, 58 will not be a factor. Also, as you mentioned, 58 is not prime. So, it's our answer! :)

May 27, 2017 • Reply

Malak Kudaimi

I'm a little confused because aren't 13, 7, 3, and 2 also not factors of 25? Also, none of the answers choices are prime numbers, so how does this explanation work?

Aug 6, 2017 • Reply


Hi Malak,

First, the question asks about 25! (25 factorial), not just 25. For more about factorials, please see this blog post:

Second, the question asks for a number that is NOT prime, so that's why none of the answer choices are prime :)

Sep 2, 2017 • Reply


Gravatar Cydney Seigerman, Magoosh Tutor

Hi Gabriela,

Happy to help :) Ok, so we're asked for the least positive integer that is not a factor of 25! and is not a prime number. Overall, we need to find the number that has a factor that is not shared by 25!

25! = 25*24*23...*2*1

The largest prime factor of 25! is therefore 23. The next prime number is 29 and 29 is the smallest prime number that is not a factor of 25!. Any multiple of 29 will also not be a factor of 25!, since any such number will always have a factor of 29. The smallest non-prime multiple of 29 is 29*2 = 58. And that's our answer :)

I hope this helps!

Jun 25, 2016 • Comment


carolyn pitta


Mar 4, 2013 • Comment


Gravatar Chris Lele, Magoosh Tutor

Oct 11, 2012 • Comment

Gabriela Perez

Is there any other way to explain this? please.

Jun 24, 2016 • Reply

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