What is the least positive integer that

Adam

That's exactly right!

Jul 8, 2019 • Reply

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

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

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

Adam

Hi Malak,

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

https://magoosh.com/gre/2011/its-not-an-excited-number-its-a-factorial/

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

Gabriela Perez

Is there any other way to explain this? please.

Jun 24, 2016 • Reply