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# What is the least positive integer that is not a factor of 30! and is

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Math Expert
Joined: 02 Sep 2009
Posts: 53065
What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 04:03
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25% (medium)

Question Stats:

71% (01:07) correct 29% (00:52) wrong based on 215 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

Kudos for a correct solution.

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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 05:20
5
Bunuel wrote:
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

Kudos for a correct solution.

Not reading carefully is going to kill me on the GMAT. My instinct was to select A because I missed the "NOT A PRIME NUMBER" part.

If you do not know what you're looking for, the best way to go about this problem is by breaking down each of the options into factors.

A) 31 is prime and per the question stem we cannot use this
B) 32 = 2 * 16 - within 30! we have a 2 and 16 so 32 will be a factor of 30!
C) 33 = 3 *11 - again within 30! we have a 3 and an 11 so 33 will be a factor of 30!
D) 62 = 31 * 2 - Bingo 31 is not included in 30!
E) 64 = 16 * 4 - both 16 and 4 are multiplied in 30!

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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 04:06
Bunuel wrote:
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

Kudos for a correct solution.

Should be 32 since 31 is a prime.
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What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 04:25
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D is the answer (62=31•2 is not a factor)

30!= 1•2•3•...11•..16•..29•30

A) 31 prime number
B) 32=16•2 factor
C) 33=11•3 factor
E)64= 16•4 factor

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What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 07:18
2
its D.

as 30 and 31 are co primes 31 will be the positive integer hat wont be a factor of 31!

but wait, question asks us that positive integer shouldnt be prime.

So find multiples of 31 as rest all other numbers in choices would be divisibleby 30! or be a prime.

therefore, 62=31*2

hope it helps
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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 19:36
Bunuel wrote:
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

Ans: D

Solution: 30! includes integer from 1-30 in multiplication form. means any value turning out to be a multiplication of this range can not be the ans and ofcourse any prime number too.
A) 31= prime number; not an answer
B) 32 = 16*2 = not an answer as 30! contains 16 and 2 in it.
C) 33 = 11*3 = not an answer
D) 62 = 31*2 = Correct as 31 is not in 30!
E) 64 = 16*4 = not an answer

Ans : D
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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 21:53
Hi SD007,

You have to be very careful with this question. The answer that you chose is the one that is "the least number greater than 30 that is NOT prime"...but it's not the answer to THIS question.

30! = (30)(29)(28)(27)....(3)(2)(1)

So we're dealing with a gigantic number that is divisible by a LOT of other numbers.

For example, we know that 30! is divisible by 60 because 30! has a (30) and a (2) in it, and (30)(2) is 60.

That same logic can help you to eliminate each of the wrong answers (besides 31, since 31 is prime and we're told to find a value that is NOT prime).

32 = (16)(2) so it too is a factor of 30! Can you eliminate the other wrong answers in the same way?

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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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14 May 2015, 22:09
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1
Celestial09 wrote:
Hi,
Couldn't understood the question yet...
waiting for the explanation and OA.
thanks
celetial

30! = 1*2*3*4*5*...*28*29*30
So all the numbers from 1 to 30 are factors of 30!. Also, all numbers which you can make by combining these factors are factors of 30!
So 2*29 = 58 is a factor of 30!, 2*4*8 = 64 is a factor of 30! and so on...

You want a number which is not a factor of 30!
The smallest number that is not a factor of 30! is 31. But it is a prime number. You want the smallest number that is not a factor of 30! and is not prime. So the next smallest such number would be 31*2 = 62.

Now think why no other number between 31 and 62 can be number which is not a factor of 30! and not prime.
32 = 4*8 (both factors of 30! so 32 is a factor of 30!)
33 = 3*11 (both factors of 30! so 33 is a factor of 30!)
34 = 2*17 (both factors of 30! so 34 is a factor of 30!)
and so on...
37 = 37 (prime)
and so on...
Basically, you want the number to be composed of two numbers such that at least one of them is not a factor of 30!. The first such number is 31. You multiply it by the smallest number 2 to get a non prime number which is not divisible by 30!.

Do focus a bit on this logic. To solve the question, you can use the options of course!
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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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15 May 2015, 16:40

31 is prime
B) 32 = 4*8 both are a factors of 30!
C) 33 = 3 *11 both are a factors of 30!
E) 64 = 16 * 4 both are a factors of 30!
so D
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Posts: 53065
What is the least positive integer that is not a factor of 30! and is  [#permalink]

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18 May 2015, 06:28
Bunuel wrote:
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

Kudos for a correct solution.

OFFICIAL SOLUTION:

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.

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Re: What is the least positive integer that is not a factor of 30! and is  [#permalink]

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Re: What is the least positive integer that is not a factor of 30! and is   [#permalink] 01 Oct 2018, 06:09
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