The function f(n) is defined as the product of all integers from 1 to

Question

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 {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

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

IanStewart · Accepted Answer

f(150) is just 150!, and when we divide 150! by all of the odd integers up to 150, we end up just with the product of the even integers from 2 to 150 inclusive. So we want to know about the smallest prime factor of

(150)(148)(146)...(6)(4)(2) + 1

If we factor out one 2 from each term in that big product, we can rewrite it more succinctly:

(150)(148)(146)...(6)(4)(2) + 1 = (2)(75)(2)(74)(2)(73)...(2)(3)(2)(2)(2)(1) + 1 = 2^75 * 75! + 1

Now, 75! is divisible by every prime less than 75. So if you take any such prime, say 47, then 2^75 * 75! will be divisible by 47, and 2^75 * 75! + 1 will be exactly 1 greater than a multiple of 47. That's another way of saying "the remainder will be 1 when we divide 2^75 * 75! + 1 by 47". So when we divide this huge number by any prime less than 75, we always get a remainder of exactly 1. Its smallest prime divisor must therefore be greater than 75.

This is, incidentally, a close copy of the famous official GMATPrep question (famous because for years it was posted to GMAT forums on a near daily basis because it was so hard) here:

https://gmatclub.com/forum/for-every-po ... 12521.html

BrushMyQuant · Answer

f(150) = The product of all integers from 1 to 150, inclusive = 1 * 2 * 3 * ... * 149 * 150 
 g(150) = The product of all odd integers from 1 to n, inclusive = 1 * 3 * 5 * ... * 149

=>  \frac{f(150)}{g(150)}  + 1 =  \frac{ 1 * 2 * 3 * ... * 149 * 150 }{ 1 * 3 * 5 * ... * 149}  + 1

Now all odd terms will cancel out and only the even terms will remain

= 2 * 4 * 6 * ... * 150 + 1
=  2^75  * (1 * 2 * ... * 75) + 1

That means when the number is divided by any number from 1 to 75 then it will give 1 as remainder
=> Number is not divisible by any number from 1 to 75
=> Any prime number which has to divide this number has to be > 75
=> p > 75

So,  Answer will be E 
Hope it helps!

Watch the following video to MASTER Functions and Custom Characters

https://www.youtube.com/watch?v=b7JoMX0gk8M

Adit_ · Answer

Bunuel  I think the question is misleading and there should be a bracket of sorts around +1 as well like:

"p is a prime factor of  "

Interpreting it without the bracket could be like 2 is a prime factor of the expression + 1, so 3 could be an answer and option E fails.

The function f(n) is defined as the product of all integers from 1 to

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The function f(n) is defined as the product of all integers from 1 to

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards