2014 AMC 8
Complete problem set with solutions and individual problem pages
The -digit numbers and are each multiples of . Which of the following could be the value of ?
- A.
- B.
- C.
- D.
- E.
Solution 1
Since both numbers are divisible by 3, the sum of their digits has to be divisible by three. . To be a multiple of , has to be either or or and so on. We add up the numerical digits in the second number; . We then add two of the selected values, to , to get . We then see that C = or and so on, otherwise the number will not be divisible by three. We then add to , to get , which shows us that C = or or and so on. To be a multiple of three, we select a few of the common numbers we got from both these equations, which could be and . However, in the answer choices, there is no or or anything greater than , but there is a , so is our answer.
Solution 2
If is divisible by , the sum of it's digits should also be divisible by . This means that or . For equation , or . Logically, you can see the correlation between our first and second equations so we can make the assumption,
