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Question

If the product of four consecutive natural numbers increased by a natural number ρ is a perfect square, then the value of ρ 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),(x+3).
Now, 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)+p
Given that this is a perfect square, the value of p=1, so that the above
equation becomes (x2+3x)2+2(x2+3x)+1=(x2+3x+1)2

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