Problem:
The 7-digit numbers 7​ 4​ A​ 5​ 2​ B​ 1​ and 3​ 2​ 6​ A​ B​ 4​ C​ are each multiples of 3. Which of the following could be the value of C?
Answer Choices:
A. 1
B. 2
C. 3
D. 5
E. 8
Solution:
For 7​ 4​ A​ 5​ 2​ B​ 1​ to be a multiple of 3, the sum 7+4+A+5+ 2+B+1=19+A+B must be a multiple of 3. Therefore A+B is 1 less than a multiple of 3. The sum 3+2+6+A+B+4+C=15+A+B+C must be a multiple of 3, so A+B+C must be a multiple of 3. Since A+B is 1 less than a multiple of 3,C must be 1 more than a multiple of 3. Only choice (A) meets this requirement.
Answer: A​.
The problems on this page are the property of the MAA's American Mathematics Competitions