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

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

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New post Updated on: 28 Mar 2019, 01:21
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Question Stats:

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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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New post 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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New post 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

Image


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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New post 31 Mar 2019, 22:07
1

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.

Answer: D
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Re: P denotes the product of all the even natural numbers upto F, and Q   [#permalink] 31 Mar 2019, 22:07
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