If n = 1 + x, where x is the product of four consecutive positive inte

Question

If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?

I. n is odd 
II. n is prime 
III. n is a perfect square

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

KanishkM · Answer

n = 1+ x

1,2,3,4 = 24 + 1 = 25
2,3,4,6 = 144 + 1= 145

Only A satisfies the above examples

Posted from my mobile device

agent1212 · Answer

x is the product of any four consecutive integers. Lets assume integers are 1,2,3,4
x=1*2*3*4=24
n=24+1=25

25 is odd as well as perfect square but its not prime...(Satisfying condition 1 and 3)
Since condition 2 is failing even in atleast 1 case we can safely say that 2 is not true.

So aption E --> I and III only should be correct

Also we can prove this algebraically :

n(n+1)(n+2)(n+3)+1 = (n^{2}+n)(n^{2}+5n+6)+1 
 = n^{4}+6n^{3}+11n^{2}+6n+1 
 = (n^{2}+3n+1)^{2} 
--> n(n+1)(n+2)(n+3) =  (n^{2}+3n+1)^{2} -1 
Since we can see that in above equation that product of any four consecutive integer is one less perfect square, and since perfect squares are even number one less than even is always odd (Satisfying condition 1 and 3)

Archit3110 · Answer

x=1,2,3,4
n= 1+24= 25
option 1 & 3 are true 
IMO E

Akshit03 · Answer

2,3,4,6 are not consecutive.
you forgot 5.

harsheyb · Answer

n is odd & a perfect square, holds for when the consecutive list is limited to (1,2,3,4) or (2,3,4,5).
n is a perfect square when we test all sequences.
So answer must be E.

GMATMBA5 · Answer

When we test (3, 4, 5, 6), the value of x is 360 (3*4*5*6) and value of n = x +1 = 361, which is square of 19.

---------------------------------------------------------------------------------------------------------------------------------
Kudos if helpful!

ScottTargetTestPrep · Answer

The product of 4 consecutive integers is always divisible by 4! = 24. Thus, n is always 1 more than a multiple of 24, so we know that n must be odd. So Roman numeral I is true.

However, n does not have to be prime. For example, if x = (1)(2)(3)(4) = 24, then n = 25, which is not a prime.

We see that n = 25 is a perfect square. Is n always going to be a perfect square? Let’s check some additional values of x.

If x = (2)(3)(4)(5) = 120, then n = 121, which is also a perfect square.

If x = (3)(4)(5)(6) = 360, then n = 361, which is also a perfect square (note: 361 = 19^2).

Thus, n will always be a perfect square.

Answer: E

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If n = 1 + x, where x is the product of four consecutive positive inte

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

If n = 1 + x, where x is the product of four consecutive positive inte

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