Question

Sum of $15$ consecutive positive integers is a perfect square. The smallest possible value of this sum is______

• A. $144$
• B. $169$
• C. $196$
• D. $225$
• E. $Noneofthese$

Answer 1

The correct option is D $225$

Let the $15$ positive integers are

$n-7,n-6,n-5,n-4,n-3,n-2,n-1,n,n+1,n+2,n+3,n+4,n+5,n+6,n+7$

Sum is: $=n-7+n-6+n-5+n-4+n-3+n-2+n-1+n+n+1+n+2+n+3+n+4+n+5+n+6+n+7=15n$

Given that $15n$ is a perfect square

$\therefore$ Smallest possible value is

$15\times 15=225$