If y=10! what is the third smallest non-prime, positive integer that is not a factor of y ?

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

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

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!

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