Digital Roots of Square Numbers Can anyone offer a proof of the following: 
The digital root of a square number is always $1$, $4$, $7$ or $9$. (It is never $2$, $3$, $5$, $6$ or $8$.) 
Digital root : Add the digits of a number until you get a single digit. 
examples: The digital root of $144$ is $1+4+4 = 9$. 
The digital root of $14289$ is $1+4+2+8+9 = 24  2+4 = 6$. 
The digital root of $1428842$ is $1+4+2+8+8+4+2 =  29  2+9 = 11 1+1 = 2$. 
Square number: A number whose square root is an integer. 
Examples: $25$    $36$    $144$    $400$ 
$116$ is not a square number. 

Mod $9$: $0^2 \equiv 3^2 \equiv 6^2 \equiv 0$, $1^2 \equiv 8^2 \equiv 1$, $2^2 \equiv 7^2 \equiv 4$, $4^2 \equiv 5^2 \equiv 7$.  That's all!
