2024 AMC 8
Complete problem set with solutions and individual problem pages
Let the letters ,,,,, represent distinct digits. Suppose is the greatest number that satisfies the equation
What is the value of ?
- A.
- B.
- C.
- D.
- E.
Solution 1
The highest that can be would have to be , and it cannot be higher than that, because then it would be , and multiplied by 8 is , and then it would exceed the - digit limit set on .
So, if we start at , we get , which would be wrong because both would be , and the numbers cannot be repeated between different letters.
If we move on to the next highest, , and multiply by , we get . All the digits are different, so would be , which is . So, the answer is .
Solution 2
Notice that .
Likewise, .
Therefore, we have the following equation:
.
Simplifying the equation gives
.
We can now use our equation to test each answer choice.
We have that , so we can find the sum:
.
So, the correct answer is .
