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Try this problem from I.S.I. B.Stat Entrance Objective Problem based on Perfect square and Positive Integer.

## Perfect Square & Positive Integers ( B.Stat Objective Question )

If n is a positive integer such that 8n+1 is a perfect square, then

• n must be odd
• 2n cannot be a perfect square
• n must be a prime number
• none of these

Perfect square

Positive Integer

Primes

## Check the Answer

But try the problem first…

Answer: 2n cannot be a perfect square.

Source

B.Stat Objective Problem 115

Challenges and Thrills of Pre-College Mathematics by University Press

## Try with Hints

First hint

Let $(8n+1)=k^{2}$ be a perfect square so k is found to be odd here as 8n+1 is odd.

Second Hint

$\Rightarrow 8n=k^{2}-1$

$\Rightarrow 8n=(k-1)(k+1)$

$\Rightarrow 2n=\frac{(k-1)(k+1)}{4}$

Final Step

$\Rightarrow (\frac{k-1}{2})(\frac{k+1}{2})$

here (k-1) and (k+1) are consecutive even numbers then $(\frac{k-1}{2})(\frac{k+1}{2})$ are consecutive even numbers

$2k \times (2k+2)= 2^2{k(k+1)}$ is not a perfect square as product of two consecutive numbers proved here as not a perfect square

So, 2n is not perfect square.