1
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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

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B

4

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C

2

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D

1

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Solution

## The correct option is 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(x2+3x)+p 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)2

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