For every positive odd integer n, the function g(n) is defined to be

Question

For every positive odd integer n, the function g(n) is defined to be the product of all the odd integers from 1 to n, inclusive. If t is the smallest prime factor of g(99) + 1, then t is?

(A) between 2 and 19
(B) between 20 and 39
(C) between 40 and 59
(D) between 60 and 79
(E) 80 or greater

I came across this question on gmatpill. Now i have seen a similar question for even integer product...how do we approach this one...?

Bunuel · Accepted Answer

g(99) = 1*3*5*..*99 = odd.

g(99) + 1 = odd + 1 = even. The smallest prime factor of any even number is 2 --> t = 2.

Answer: A.

Bunuel · Answer

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LighthousePrep · Answer

This question tries to trick you into thinking that you have to find the prime factorization of a very huge number. In fact, you won't have to do that at all.

Consider what g(99) means:
g(99) = 1 * 3 * 5 * 7 * 9 * ... * 97 * 99 = some huge number.

You can already establish that the smallest prime factor of that huge number is 3, since it is defined within what g(x) means.

Since t = smallest prime factor (which will always be 3 when n > 1) + 1, we can conclude that t = 4.

The correct answer is A.

TheDelhiDude · Answer

Thanks Bunuel so its safe to assume if the question involves a product of consecutive odd integers (from any odd no. till any odd no.) and adds 1 to the product, it can be said that the smallest prime factor is 2.

Say if its g(n)=[5x7x9x...any no.]+1 , the smallest prime factor is 2?

Say if its g(n)= +1 , the smallest prime factor is 2?

Bunuel · Answer

Well, yes. How else?

The product of odd numbers (consecutive or not) is odd. ODD + 1 = EVEN. The smallest prime factor of any even number is 2.

LighthousePrep · Answer

Thanks for the clarification Bunuel, but maybe there could be two interpretations of the question?

My understanding was that the question was defining t =   + 1, rather than t = smallest prime factor of (g(99)+1).

I thought that because answer choice A did not clarify "between 2 and 19 INCLUSIVE" (which the GMAT typically does, right?).

Either way, the answer choice is A.

TeamGMATIFY · Answer

The key here is not to get intimidated by the huge number g(99)

g(99) = 1*3*5* ...*97*99
We need to find the smallest prime factor of g(99) + 1

We know that:
 Odd*Odd = Odd
Odd*Even = Even
Even*Even = Even
 
Hence g(99) = 1*3*5* ...*97*99 will always be an odd number

We also know that
 Odd + Odd = Even
Even + Even = Even
Odd + Even = Odd

Therefore g(99) + 1 has to be an even number
 The smallest prime factor of any even number = 2 
Hence Option A

itzmyzone911 · Answer

Bunuel: One doubt?

What if the question were reworded to find the largest prime number of g(99)+1?

Bunuel · Answer

GMAT would never ask such question because the largest prime of g(99) +1 is 7728802419974390867942174885150693176068821337346448420812666105479729917, which is not that easy to find.

itzmyzone911 · Answer

Wow what a number! :roll:

Just for curiosity, Bunuel is this a symbolic representation (fictitious) of the prime factor or you actually calculated it. Can you explain how you arrived at this no?...and in any case if the options were kept the same, option E would have been the correct answer?

Bunuel · Answer

This is the largest prime of g(99) + 1. I used special calculator to get it.

itzmyzone911 · Answer

Ok...thanks Bunuel :-D

karim2982 · Answer

I agree that t=2, but the answer choice A reads: 2 < t < 19. Doesn't that make t=2 the wrong answer, since t={3,4,...,18}?
Can you please clarify?

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For every positive odd integer n, the function g(n) is defined to be

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