Spring 2010 Problem 4: Adding two hundred

There are two two-digit numbers ab such that 1) if you put a 2 in front of ab, the result is evenly divisible by ab (so ab divides 200+ab), and 2) if you add 20 to ab, the result evenly divides 200+ab. Find these two numbers.

Solution

As Dr. Goadrich pointed out, there are actually three solutions: 10, 40, and 25. (I only caught 10 and 40.) Dallas Krenztel answered first (getting two points for the answer and one point for the first correct answer), while Josh Pena earned two points for the correct answer. Scott Jackson also submitted one of the correct numbers.

By the way, a good way to narrow down the numbers is as follows: if ab evenly divides 200+ab, then ab evenly divides 200 (since it already divides ab). So you only need to focus on the divisors of 200.

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Solution