1997 PMWC Problems/Problem I2

Problem

In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number stand for?

\[\begin{tabular}{c c c c c}& &\Box & 1 &\Box\\ &\times & & 9 &\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{tabular}\] (Error compiling LaTeX. Unknown error_msg)

Solution

Following the rules of multiplication, we see that 9 times the units digit of the three digit number ends in 7, which means that the digit must be a 3. Carrying out the multiplication, we see that the last two digits of the second product are 17, which means that the hundreds digit in the first product must be a 4. We now have

   -13
   *9-
   ___
  -49-
 --17
 _____
 HA66Y

The only digit that would work as the units digit of 9- is 7. Therefore we have

   -13
   *97
   ___
  -491
 --17
 _____
 HA661

The only multiple of 7 that is two digits and is 7 times a digit is 14. Therefore we have

   213
   *97
   ___
  1491
 1917
 _____
 20661