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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

7 Explanations


chen chen


26= 2*13
28=2*14or 4*7
36=2*18or 3*12 or 4*9
All products above are inside 25! So they are all factors of 25!
But 58=2*29 and 29 is not inside 25! so it's not a factor of 25!

Jun 27, 2019 • Comment


That's exactly right!

Jul 8, 2019 • Reply


Michelle Scire

So why doesn't the 2 count as a factor of 25? Thats what I am confused on.

Apr 9, 2018 • Comment

Sam Kinsman

Hi Michelle!

Actually, 2 is a factor of 25! Keep in mind that 25! = 1*2*3*4*5.....*24*25. This list of numbers is divisible by 2, so therefore 2 is a factor of 25!

When we find that 29 is the smallest prime number that is greater than 25, we multiply 29 by 2 (because the answer cannot be a prime number, so 29 itself doesn't work). We get 58. You're right that 58 is divisible by 2, and that 2 is a factor of 25 - but that doesn't matter here, because even though 2 is a factor of 25, the number 58 is not a factor of 25!. In other words, 25! is not divisible by 58. So that's why 58 is the number we're looking for :)

Apr 11, 2018 • Reply


I wonder the passage.
I thought passage say 'not a factor of 25! and not a prime number'. so I thought 26 is also correct. because 26 is not a factor (yes 2*13 is factor of 25! but 26 is more than 25!. so I thought only 26, not 2 and 13) and it's not prime number.

If the answer was 58 like explanation from other, passage should say like this,
"the least positive integer that's factor is not a factor of 25!"

I thought It's about English Expression, am I wrong? what part of passage say number's factor is not a factor of 25! not just hall number?

Mar 2, 2018 • Comment

David Recine

This isn't a matter of the English wording exactly. Instead, what you need to understand here is that even if a number is greater than 25 (without the exclamation point), it could still be a factor of 25! (25 with the exclamation point at the end, or 25 factorial).

Bear in mind that 25! is actually a very big number. It is the product of 25, multiplied by every positive whole number that is less than 25. The ultimate product is a number that is over 25 digits long. As a result, many very large numbers, including 26 and numbers in the hundreds, thousands, millions, billions, etc... could be factors of 25!

For a number to not be a factor of 25!, it must be larger than 25! AND also not be a product of the numbers that are multiplied by each other to get the product of 25!

Since 25! is 25 multiplied by every whole positive number less than 25, it includes 13 multiplied by 2, which equals 26. In other words, if 25! can be divided evenly by 13*2, then it can be divided by 26.

For this reason, 27 (9*3), 28 (7*4), 36 (9*4), and 56 (14*4) also are factors of 25! A number being larger than 25 does not automatically make it NOT a factor of 25!. Instead, a number larger than 25 is only *possibly* not a factor of 25! You still have to check any 25+ number to see if its prime factorization consists entirely of factors that can be found within 25!

Mar 4, 2018 • Reply

Thank for you clear answer ! I understood what you mean !

Mar 6, 2018 • Reply


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


Chris Lele

Oct 11, 2012 • Comment

Gabriela Perez

Is there any other way to explain this? please.

Jun 24, 2016 • Reply

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