A number has two digits. If the digits are interchanged and the new number is added to the original, the result is divisible by what?

Problem restatementLet the original number be 10a + b. After swapping digits we get 10b + a. Show that their sum is always divisible by a fixed number.

Concept/ApproachAlgebraic expression of the sum reveals a common factor.

Step-by-Step calculation(10a + b) + (10b + a) = 11(a + b)Thus the sum is a multiple of 11

Common pitfallsGuessing 9 (digit-sum test) or 10 (last digit 0) without algebra; neither is generally true here.

Final Answer11