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

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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.

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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

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