If y=10! what is the third smallest non-prime, positive integer that is not a factor of y ?
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34
Is this correct?
Edit: Nvm. The correct ans is 33 and proved below.
10! =1x2x3x**4(2x2)x5x6(2x3)x7x8(2x2x2)x9(3x3)x10(2x5). Through which let us find non-prime factors first. Start from the smallest, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, … Okay, now we could recognize this pattern, seems like all the numbers within 1-10 and their multiples fall into this category apart from 11,13,17,19 and their multiples, so the first smallest, non-prime number is 22, the second is 26, the third one would be 33
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10!= 2^8 x 3^4 x 5^2 x 7. -----> this can also be skip if u can remember the fundamental theorem of arithmetic
So, intuitively start with multiples of primes that are not included it the above set, i.e, 11,13 ,17 etc (Because multiple of a prime is not a prime) as the fundamental theorem of arithmetic says every positive number greater than 1 can be expressed a product of prime numbers and as 10! covered primes till 2 ,3,5,7 ----> You should start with primes after that i.e 11,13… and the answer will be one of their multiples.
So question becomes as 10! covers PRIMES(2,3,5,7) --> we start looking at PRIMES (11,13,17…) and their multiples as question asked third smallest we start will multiples of 11 (11x2=22,11x3=33) and 13(13x2=26) …So 33
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i thought the third smallest would be 2x17=34 but didnt consider 3x11…
If y=10!, what is the third smallest non-prime, positive integer that is not a factor of y?
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
The three smallest numbers that would not be factors of the above are prime numbers > 10, i.e., 11,13,17.
However, the questions asks for non-primes. So, the three smallest would be -
11 * 2 =22
13 * 2 = 26
11 * 3 = 33 ← Answer
P.S. 17 * 2 = 34 is a trap! 
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oh my gosh, I fell into the trap and could NOT think of multiplying by 3…
Thank you!
Isn’t 25 a non prime number? So shouldn’t the answer be 26?
Since 10! consists of 5 and 10, 25 gets cancelled out.
(50 is a multiple of 25)
Thanks man, hope I don’t do these mistakes on the real GRE 
I believe the multiples of prime numbers will be the answer.
Like 11,13,17,19 are prime numbers that are not the factors of 10!
So multiplying them with 2 should give us non primes.
11x2-22 (1st)
13x2-26 (2nd)
17x2-34 (3rd)
19x2-38
but when I try 34 as the answer, it is wrong. Can anyone help me with this question?
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Small mistake, 11*3 = 33, which is non-prime and less than 34
22,26,33,34,39… so on