987654321 squared. Worked on a 8-digit 4 function inexpensive simple calculator

We can rewrite the number $987654321$ as an arithmetic-geometric progression as follows:

$\begin{aligned} 987654321 & = \sum_{n=0}^8 10^n(n+1) = \frac {8\times 10^{10}+1}{9^2} \\ \implies 987654321^2 & = \frac {64 \times 10^{20}+16\times 10^{10}+1}{9^4} \\ & = \frac {6,400,000,000,160,000,000,001}{9^4} & \small \red{\text{And using paper and pencil to}} \\ & = \frac {711,111,111,128,888,888,889}{9^3} & \small \red{\text{do the long division carefully.}} \\ & = \frac {79,012,345,680,987,654,321}{9^2} & \small \red{\text{No calculator needed.}} \\ & = \frac {8,779,149,520,109,739,369}9 \\ & = \boxed{975,461,057,789,971,041} \end{aligned}$

This method may be considered cheating.

Through the divisibility test for $9$ , we know that $987654321$ is divisible by $9$ . Therefore, we can use the divisibility test for $9$ on each of the answer responses, and only $\boxed{975461057789971041}$ , or the last answer choice, satisfies the divisibility test. (Performing the divisibility test accurately is left as an exercise to the reader.)

Since one out of the four answers is right, it has to be that one.

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

$\begin{array}{r|rrr} \text{times}& 9 & 8765 & 4321 \\ \hline 9 & 81 & 78885 & 38889 \\ 8765 & 78885 & 76825225 & 37873565 \\ 4321 & 38889 & 37873565 & 18671041 \\ \end{array}$

Then, extend the numbers with the appropriate number of zeroes and add down the columns:

$\begin{array}{r|rrr} & 9 & 8765 & 4321 \\ \hline 9 & 810000000000000000 & 78885000000000000 & 3888900000000 \\ 8765 & 78885000000000000 & 7682522500000000 & 378735650000 \\ 4321 & 3888900000000 & 378735650000 & 18671041 \\ \hline + & 888888888900000000 & 86567901235650000 & 4267654321041 \end{array}$

Then, add the column sums: $\begin{array}{r} 888888888900000000 \\ 86567901235650000 \\ 4267654321041 \\ \hline 975461057789971041\end{array}$