Square Total

Answer: $5$


Anastasia: $a$
    Barry:   $a\times2=2a$
  Connor:  $2a\times3=6a$
  Damion: $6a\times6=36a$
      Eve:   $a+2a+6a+36a=45a$ is a square number.


Using the $45$ times table

$a$	$45a$	Square?
$1$	$45$	No
$2$	$90$	No
$3$	$135$	No
$4$	$180$	No
$5$	$225$	Yes: $15^2 = 225$

This shows that the smallest possible number would be $5$.


Alternatively, for $45a$ to be a square number, each of its prime factors must be raised to an even power. Since $45 = 3^2\times 5$ as a product of prime factors, the only prime factor not raised to an even power is $5$. Therefore the smallest value of $a$ that makes this a square must be $5$.