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.
See also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 60 |
Followed by: Problem TB2 | |
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