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

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

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New post 16 Nov 2018, 04:08
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E

Difficulty:

  45% (medium)

Question Stats:

66% (02:07) correct 34% (02:33) wrong based on 115 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

    A. p < 10
    B. 10 < p < 25
    C. 25 < p < 50
    D. 50 < p < 75
    E. p > 75

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

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New post 16 Nov 2018, 07:40
2
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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Re: Question of the Week- 23 (The function f(n) is defined as the .......)  [#permalink]

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New post 16 Nov 2018, 07:58
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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New post 16 Nov 2018, 08:07
keatross wrote:
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.


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

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New post 28 Nov 2018, 06:31

Solution


Given:
    • f(n) = 1 * 2 * 3 * 4 * … * (n - 1) * n
    • g(n) = 1 * 3 * 5 * 7 * … * (n - 1), if n is even, or g(n) = 1 * 3 * 5 * 7 * … * n, if n is odd
    • p is a prime factor of \(\frac{f(150)}{g(150)} + 1\)

To find:
    • Which of the given options is always true?

Approach and Working:
    • f(150) = 1 * 2 * 3 * 4 * … * 149 * 150
    • g(150) = 1 * 3 * 5 * 7 * … * 149
    • Implies, \(\frac{f(150)}{g(150)} = 2 * 4 * 6 * 8 * … * 148 * 150 = 2^{75} * (1 * 2 * 3 * 4 * … * 75) = 2^{75} * 75!\)
      o Let us assume that \(2^{75} * 75! = N\)

    • So, \(\frac{f(150)}{g(150)} + 1 = 2^{75} * 75! + 1 = N + 1\)
    • Now, we know that any two consecutive integers are co-primes
      o Thus, N and N + 1 are co-primes
      o If we observe, \(N = 2^{75} * 75!\), which is a multiple of all the primes from 1 to 75,
      o So, all the prime factors from 1 to 75 are not the factors of N + 1, as N and N + 1 are co-primes

    • Therefore, we can say that the prime factor of N + 1 must be greater than 75

Hence the correct answer is Option E.

Answer: E

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Re: Question of the Week- 23 (The function f(n) is defined as the .......)   [#permalink] 28 Nov 2018, 06:31
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