\($ = \sqrt{@}\)
\(& = @ + $\)

\(=> & = @ + \sqrt{@}\)

Thus, \(&\) is equal to the sum of a number and that number's square-root

Since the numbers are positive integers, \(@\) must be a perfect square

Possible values of \(@\) are:

\(@\) = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
\(\sqrt{@}\) = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

\(&\) = 2, 6, 12, 20, 30, 42, 56, 72, 90, 110


We need to still verify Option F:
289 + \sqrt{289} = 306
324 + \sqrt{324} = 342
Thus, 331 cannot be expressed.


[b]An Alternate approach could have been:
\(=> & = @ + \sqrt{@}\)
\(=> & = \sqrt{@} * (\sqrt{@} + 1)\)

Since the numbers are positive integers, \(@\) must be a perfect square
Thus, \(&\) is the product of 2 consecutive integers. Only 30 = 5 * 6 and 72 = 8 * 9 satisfy


Thus, Options B and D are possible.