Question of the Week- 23 (The function f(n) is defined as the .......) : Problem Solving (PS)

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Question of the Week- 23 (The function f(n) is defined as the .......)

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e-GMAT Representative

Joined: 04 Jan 2015

Posts: 2326

Question of the Week- 23 (The function f(n) is defined as the .......) [#permalink]

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16 Nov 2018, 03:08

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Difficulty:

55% (hard)

Question Stats:

59% (01:42) correct

41% (02:05) wrong

based on 91 sessions

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e-GMAT Question of the Week #23

The function f(n) is defined as the product of all integers from 1 to n, inclusive, and the function g(n) is defined as the product of all odd integers from 1 to n, inclusive, where n is a positive integer. If p is a prime factor of \(\frac{f(150)}{g(150)} + 1\), then which of the following must be true

Spoiler: OA

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Joined: 14 Jun 2018

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Question of the Week- 23 (The function f(n) is defined as the .......) [#permalink]

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16 Nov 2018, 06:40

Let k = f(n)/g(n) + 1
product of all integers / product of odd integers = product of even integers

Question is asking us to find the product of all even integers between 1 and 150.

k = 2x4x6x...150.
k = 2^75(1*2*3*...75)

Now k+1 and k are co prime (two consecutive no's are co prime) . So k+1 won't have any factors between 2-75 inclusive

So p > 75

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Joined: 15 Nov 2018

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Re: Question of the Week- 23 (The function f(n) is defined as the .......) [#permalink]

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16 Nov 2018, 06:58

Going to need some help on this one for sure (and I do not understand the solution posted)....

I tried to find a pattern between the sizes of f(n) and g(n) given their properties. Didn't really result in much I could use.

I did, however, notice that g(n) will have to skip half of the #s from 1-150, which is 75. Using this I kind of theorized that f(n) will be 75(x all even numbers) times larger than g(n). [E] was my guess as a result, but it was a complete shot in the dark.

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Question of the Week- 23 (The function f(n) is defined as the .......) [#permalink]

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16 Nov 2018, 07:07

keatross wrote:

i hope this will be clearer

f(150) = 1x2x3x4x5x....150 (product of all integers between 1-150 inclusive)
g(150) = 1x3x5x7x9...149 (product of all odd integers between 1-150)

\(\frac{f(150)}{g(150)}\) = 2x4x6x8x10....150

=> 2^75 (1*2*3*4*...75) [take 2 common from all the terms. There are 75 terms therefore 2^75]

Prime Factor for \(\frac{f(150)}{g(150)}\) = all prime factors from 1 to 75

Since two consecutive terms have no prime factors common , Prime Factor for \(\frac{f(150)}{g(150)}+ 1\) will not have any term between 1 and 75

Therefore p>75