1997 AHSME Problems/Problem 1

Problem

If $\texttt{a}$ and $\texttt{b}$ are digits for which

$\begin{tabular}{ccc}& 2 & a\\ \times & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular}$ (Error compiling LaTeX. Unknown error_msg)

then $\texttt{a+b =}$

$\mathrm{(A)\ } 3 \qquad \mathrm{(B) \ }4 \qquad \mathrm{(C) \ } 7 \qquad \mathrm{(D) \ } 9 \qquad \mathrm{(E) \ }12$

Solution

From the units digit calculation, we see that the units digit of $a\times 3$ is $9$ . Since $0 \le a \le 9$ and $a$ is an integer, the only value of $a$ that works is is $a=3$ . As a double-check, that does work, since $23 \times 3 = 69$ , which is the first line of the multiplication.

The second line of the multiplication can be found by doing the multiplication $23\times b = 92$ . Dividing both sides by $23$ gives $b=4$ .

Thus, $a + b = 3 + 4 = 7$ , and the answer is $\boxed{C}$ .

1997 AHSME (Problems • Answer Key • Resources)
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1997 AHSME Problems/Problem 1

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