982451653 squared. Worked on a 8-digit 4 function inexpensive simple calculator

The Digital Root Test saves the day again!

We know that $982451653$ is congruent to $7 \mod 9$ . So $982451653^2$ is congruent to $49\mod 9$ , or $4\mod 9$ .

Checking all the answers, we find that only $\boxed{965211250482432409}$ is congruent to $4\mod 9$ .

Write the number $987654321$ as a base 1000 integer 9, 8765, and 4321.

Generate the appropriate digit products, noting the number of occurrences of that product with the proper number of low-order zeroes:

$\begin{array}{rr|r} \text{extended product} & \text{\#} & \text{result} \\ \hline \text{81,0000,0000,0000,0000} & 1 & \text{81,0000,0000,0000,0000} \\ \text{7,4205,0000,0000,0000} & 2 & \text{14,8410,0000,0000,0000} \\ \text{1,4877,0000,0000} & 2 & \text{2,9754,0000,0000} \\ \text{6798,0025,0000,0000} & 1 & \text{6798,0025,0000,0000} \\ \text{1362,8985,0000} & 2 & \text{2725,7970,0000} \\ \text{273,2409} & 1 & \text{273,2409} \\ \hline & & \text{96,5211,2504,8243,2409} \\ \end{array}$