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M31-11

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Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139513 [0], given: 12794

M31-11 [#permalink]

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09 Jun 2015, 06:49
00:00

Difficulty:

35% (medium)

Question Stats:

72% (00:42) correct 28% (00:39) wrong based on 25 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
[Reveal] Spoiler: OA

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Kudos [?]: 139513 [0], given: 12794

Math Expert
Joined: 02 Sep 2009
Posts: 43335

Kudos [?]: 139513 [0], given: 12794

Re M31-11 [#permalink]

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09 Jun 2015, 06:50
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!.

Answer: D
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Re: M31-11 [#permalink]

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29 Dec 2017, 14:35
Tricky dicky

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Re: M31-11   [#permalink] 29 Dec 2017, 14:35
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