Track / Stage 3 / 25 of 260 #25 of 2000

Problem 25

AMC 10/12, early questions

Combinatorics Difficulty 3.1 Find the answer

The addition below is incorrect. The display can be made correct by changing one digit $d$ , wherever it occurs, to another digit $e$ . Find the sum of $d$ and $e$ .
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$
$\mathrm{(A) \ 4 } \qquad \mathrm{(B) \ 6 } \qquad \mathrm{(C) \ 8 } \qquad \mathrm{(D) \ 10 } \qquad \mathrm{(E) \ \text{more than 10} }$

Your answer

Official solution

If we change $0$ , the units column would be incorrect.
If we change $1$ , then the leading $1$ in the sum would be incorrect.
However, looking at the $2$ in the hundred-thousands column, it would be possible to change the $2$ to either a $5$ (no carry) or a $6$ (carry) to create a correct statement.
Changing the $2$ to a $5$ would give $745586 + 859430$ on top, which equals $1605016$ . This does not match up to the bottom.
Changing the $2$ to a $6$ gives $746586 + 869430$ on top, which has a sum of $1616016$ . This is the number on the bottom if the $2$ s were changed to $6$ s.
Thus $d=2$ and $e=6$ . so $d+e= 8 \boxed{\mathrm{ (C)}}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.