1997 AIME Problems/Problem 3

Problem

Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?

Solution

Let the two-digit number be ab, and the three-digit number be cde.

$abcde=10000a+1000b+100c+10d+e=9(10a+b)(100c+10d+e}$ (Error compiling LaTeX. Unknown error_msg)

9y-1000 must be positive, so y>111.

Since x is greater than 89, we can try certain factors of 1000:

100: 9x=101, nope.

125: 9x=126, x=14

Then 9y-1000=8, 1008=9y, y=112.

112+14=126