# 2009 AMC 10A Problems/Problem 5

## Problem

What is the sum of the digits of the square of $\text 111,111,111$? $\mathrm{(A)}\ 18\qquad\mathrm{(B)}\ 27\qquad\mathrm{(C)}\ 45\qquad\mathrm{(D)}\ 63\qquad\mathrm{(E)}\ 81$

## Solution 1

Using the standard multiplication algorithm, $111,111,111^2=12,345,678,987,654,321,$ whose digit sum is $81\longrightarrow \fbox{(E)}.$ (I hope you didn't seriously multiply it outright...)

## Solution 2

Note that $11^2 = 121 \\ 111^2 = 12321 \\ 1111^2 = 1234321.$

We observe the pattern and use it to find that $111,111,111^2=12,345,678,987,654,321$ whose digit sum is $81\longrightarrow \fbox{(E)}.$

## Solution 3

You can see that $111*111$ can be written as $111+1110+11100$, which is $12321$. We can apply the same fact into 111,111,111, receiving $111111111+1111111110+11111111100... = 12,345,678,987,654,321$ whose digits sum up to $81\longrightarrow \fbox{E}.$

## Solution 4

We can also do something a little bit more clever rather than just adding up the digits though. Realize too that $1^2 = 1$ $11^2 = 121$ $111^2 = 12321$ $1,111^2 = 1234321$

We clearly see the pattern, the number of digits determines how high the number goes, as with $111^2$, it has $3$ digits so it goes up to $1,2,3$ then decreases back down. If we start adding up the digits, we see that the first one is $1$, the second is $2 + 1 + 1 = 4$, the third one is $1 + 2 + 3 + 2 + 1 = 9$, and the fourth one is $16$. We instantly see a pattern and find that these are all square numbers. If the number you square has $4$ digits, you do $4^2$ to see what the added digits of that particular square will be. In this case, we are dealing with $111,111,111$ which has $9$ digits so $9^2$ equals $81\longrightarrow \fbox{E}$

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