PROBLEM 1-9-4

A three-digit number is reversed, so that the units digit and hundreds digit are interchanged, to form a larger number. The product of the two numbers is 65 125. What was the original number?

SOLUTION 1-9-4

125

Since the units digit of the product is 5, one of the two numbers has a units digit of 5 and the other number has a hundreds digit of 5. Since 500 x 200 = 100 000, which is greater than 65 125, the smaller number nust be less than 200. It therefore has a hundreds digit of 1 and a ones digit of 5. Letting b be the tens digit of both numbers, the original number has didits 1b5 and its “reverse” has digits 5b1. Examining the results of using one of the standard multiplication algorithms on the product, as illustrated below, we see that since e is the units digit of a multiple of 5 (namely, b x 5), it must equal either 0 or 5. If e were 5, then b would have to be 7, since the units digit of e + b is 2. But 175 x 571 = 99 925, not 65 125, so e = 0, which implies that b= 2. Since 125 x 251 = 65 125, we see that the original number was 125.

1b 5 x 5b 1
=1b 5
cde
fg 5
=65 125