2007 iTest Problems/Problem TB1

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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 $N$ 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.

See also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 60
Followed by:
Problem TB2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 TB1 TB2 TB3 TB4