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# P denotes the product of all the even natural numbers upto F, and Q

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3078
P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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Updated on: 28 Mar 2019, 01:21
2
2
00:00

Difficulty:

55% (hard)

Question Stats:

61% (02:27) correct 39% (02:34) wrong based on 80 sessions

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P denotes the product of all the even natural numbers upto F, and Q denotes the product of all the odd natural numbers upto F. What is the minimum value F can have, if the product of P and Q is a multiple of 3213?

A. 7
B. 9
C. 13
D. 17
E. 27

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Originally posted by EgmatQuantExpert on 27 Mar 2019, 22:57.
Last edited by EgmatQuantExpert on 28 Mar 2019, 01:21, edited 1 time in total.
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Re: P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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27 Mar 2019, 23:12
2
Given in question,

P denotes the product of all the even natural numbers upto F, and Q denotes the product of all the odd natural numbers upto F.

This means product of P and Q gives the product of all natural numbers upto F.

So product of all natural numbers upto F is a multiple of 3213.

3213 = 3*3*3*7*17

From above product of prime factors we can conclude that the product of all natural numbers till 17 results in a multiple of 3213.

OA: (D)
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Re: P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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29 Mar 2019, 02:01
EgmatQuantExpert wrote:
P denotes the product of all the even natural numbers upto F, and Q denotes the product of all the odd natural numbers upto F. What is the minimum value F can have, if the product of P and Q is a multiple of 3213?

A. 7
B. 9
C. 13
D. 17
E. 27

factorize 3213 ; 3^3*7*17
min value of F has to be 17 ; to be a multiple of 3213
IMO D
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Re: P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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31 Mar 2019, 22:07
2

Solution

Given:
• P = product of all the even natural numbers upto F.
• Q = product of all the odd natural numbers upto F.
• The product of P and Q is a multiple of 3213.

To find:
• The minimum value of F.

Approach and Working:
As per the definition of P and Q, P is the product of all even natural numbers upto F and Q is the product of all odd natural numbers upto F.
• Therefore, P * Q = product of all the natural numbers upto F = Factorial of F.
• So, as per the given question, F! is a multiple of 3213.

Now, if we express 3213 in terms of its prime factors, we get $$3213 = 3^3 * 7 * 17$$

Therefore, in F!, we will have $$3^3, 7$$ and $$17$$ at least.

So, the minimum value of F must be 17.

Hence, the correct answer is option D.

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Re: P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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26 Apr 2019, 08:07
What is the easiest way to find the prime factors of such a large number? I took a very long time to solve this problem and it probably shouldn't have taken me this long...

Any tips or tricks would be much appreciated! Thank you in advance!
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Re: P denotes the product of all the even natural numbers upto F, and Q  [#permalink]

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17 Jun 2019, 05:46
jojo95 wrote:
What is the easiest way to find the prime factors of such a large number? I took a very long time to solve this problem and it probably shouldn't have taken me this long...

Any tips or tricks would be much appreciated! Thank you in advance!

Hey,
There are no tricks to prime factorisation or at least I'm not aware of any. Irrespective, the given number is not hard to prime factorise.
To tell you the truth, when I started solving the question myself, my first thought on seeing the number was there is no way GMAT will ask me to factorise this. There has to be a better way.
After wasting a minute and failing to come up with an alternate method, I did what was needed. PRIME FACTORISATION! and to my surprise it was really easy. The only thing holding me back was my fear/thought.

I don't know why you took a really long time to solve this. If you were stuck on where to start factorising - JUST PICK A NUMBER. Start somewhere, move on, repeat!

Hope I helped you in some way.
Re: P denotes the product of all the even natural numbers upto F, and Q   [#permalink] 17 Jun 2019, 05:46
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