Question

If the product of four consecutive natural numbers increased by a natural number p is a perfect square, then the value of p is 

(a) 8 

(b) 4 

(c) 2 

(d) 1

Answer 1

(d) 1

Let the four consecutive natural numbers be 

x, x +1, x + 2 and x + 3. Then,

A perfect square= x (x + 1) (x +2) (x + 3) + p 

= x (x + 3) (x + 1) (x + 2) + p 

= (x² + 3x) × (x² + 3x + 2) + p 

= (x² + 3x)² + 2 (x² + 3x + 2) + p 

= (x² + 3x)² + 2(x 2 + 3x) + p T

The expression on the right hand side will be a perfect square if and only p = 1.

Perfect square number 

= [(x² + 3x)² +2 (x² + 3x) + 1] 

= (x² + 3x +1)