2007 iTest Problems/Problem TB1

Problem

The sum of the digits of an integer is equal to the sum of the digits of three times that integer. Prove that the integer is a multiple of 9.

Solution

A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively. A proof can be found here. Since the sum of the digits of 3 times the integer is a multiple of 3, the sum of the digits of the original integer is a multiple of 3, and hence the original integer is a multiple of 3. The sum of the digits of 3 times the integer is a multiple of 9. Hence the sum of the digits of the number itself is a multiple of 9. Hence the integer itself is a multiple of 9.