   SEARCH HOME Math Central Quandaries & Queries  Question from Audrey, a student: the sum of the digits of two-digit number is 9. if the digits are reversed, the new number if 63 greater than the original number. find the number. Hi Audrey,

You can certainly use algebra to solve this problem.

Suppose the digits are m and n so the number is 10m + n. You know two facts

The sum of the digits is 9

m + n = 9

If the digits are reversed, the new number if 63 greater than the original number.

10n + m = 10m + n + 63

Solve these two equations for m and n.

There is however an easier and more insightful way to solve the problem. It involves knowing one fact.

If the sum of the digits of a number s divisible by 9 then the number is divisible by 9.

The sum of the digits of the two-digit number you started with is 9 which is divisible by 9. Hence the two-digit number is divisible by 9. What are the two-digit numbers which are divisible by 9?

2 × 9 = 18
3 × 9 = 27
4 × 9 = 36
.
.
.
11 × 9 = 99

Add 63 to each of these. Which one results in a two-digit number with the digits reversed?

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.