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Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Difficulty:   35% (medium)

Question Stats: 78% (00:50) correct 22% (00:33) wrong based on 36 sessions

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What is the least positive integer that is not a factor of 30! and is not a prime number?

A. 31
B. 32
C. 33
D. 62
E. 64

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Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Official Solution:

What is the least positive integer that is not a factor of 30! and is not a prime number?

A. 31
B. 32
C. 33
D. 62
E. 64

We need such number which is not a factor of 30! and is not a prime number.

The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is therefore 2*31=62. Notice that all numbers between 30 and 62, are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all their primes in higher powers). For example:

31 is a prime, hence x cannot be 31.

$$32 = 2^5$$. 30! will for sure have 2 in higher power than 5, hence 32 IS a factor of 30!.

$$33 = 3*11$$. Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!

...

$$64= 2^6$$. 30! will for sure have 2 in higher power than 6, hence 64 IS a factor of 30!.

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Intern  B
Joined: 05 Feb 2017
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Tricky dicky
Manager  G
Joined: 16 May 2016
Posts: 198
Location: India
Schools: ESSEC '21 (A\$)
GMAT 1: 720 Q50 V38 GPA: 3.5
WE: Analyst (Consulting)

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Bunuel wrote:
Official Solution:

What is the least positive integer that is not a factor of 30! and is not a prime number?

A. 31
B. 32
C. 33
D. 62
E. 64

We need such number which is not a factor of 30! and is not a prime number.

The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is therefore 2*31=62. Notice that all numbers between 30 and 62, are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all their primes in higher powers). For example:

31 is a prime, hence x cannot be 31.

$$32 = 2^5$$. 30! will for sure have 2 in higher power than 5, hence 32 IS a factor of 30!.

$$33 = 3*11$$. Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!

...

$$64= 2^6$$. 30! will for sure have 2 in higher power than 6, hence 64 IS a factor of 30!.

I somehow did not understand this explanation.
Math Expert V
Joined: 02 Sep 2009
Posts: 59674

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Cbirole wrote:
Bunuel wrote:
Official Solution:

What is the least positive integer that is not a factor of 30! and is not a prime number?

A. 31
B. 32
C. 33
D. 62
E. 64

We need such number which is not a factor of 30! and is not a prime number.

The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is therefore 2*31=62. Notice that all numbers between 30 and 62, are either primes (and we know that x is NOT a prime) or factors of 30! (because 30! has all their primes in higher powers). For example:

31 is a prime, hence x cannot be 31.

$$32 = 2^5$$. 30! will for sure have 2 in higher power than 5, hence 32 IS a factor of 30!.

$$33 = 3*11$$. Both 3 and 11 are factors of 30!, hence 33 IS a factor of 30!

...

$$64= 2^6$$. 30! will for sure have 2 in higher power than 6, hence 64 IS a factor of 30!.

I somehow did not understand this explanation.

We need such number which is not a factor of 30! and is not a prime number.

The smallest prime which is not a factor of 30! is obviously 31. So, the smallest positive integer that is not a factor of 30! and is not a prime number is 2*31=62.

Check more solutions here: https://gmatclub.com/forum/what-is-the- ... 98237.html

Hope it helps.
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LBS Moderator V
Joined: 04 Jun 2018
Posts: 664
Location: Germany
Concentration: General Management, Finance
GMAT 1: 730 Q47 V44 GPA: 3.4
WE: Analyst (Transportation)

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The fact that we are directly told to look for a none prime number does give us the key to solving this question.

As we know 31 is a prime number, we can look for the next smallest answer choice that does contain a prime factor which is not part of 30!.
Once we see that 62 = 2x31 we know we have found our right answer.

Best regards,
Chris
_________________ Re: M31-11   [#permalink] 27 Sep 2018, 03:59
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# M31-11

Moderators: chetan2u, Bunuel  