Difference between revisions of "2024 AMC 8 Problems/Problem 15"
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Revision as of 15:00, 25 January 2024
Problem
Let the letters ,,,,, represent distinct digits. Suppose is the greatest number that satisfies the equation
What is the value of ?
Solution 1
The highest that FLYFLY can be would have to be 124124, and cannot exceed that because it would exceed the 6-digit limit set on BUGBUG.
So, if we start at 124124*8, we get 992992, which would be wrong because the numbers cannot be repeated.
If we move on to 123123 and multiply by 8, we get 984984, all the digits are different, so FLY+BUG would be 123+984, which is 1107. So, therefore, the answer is C, 1107.
-Akhil Ravuri, John Adams Middle School
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 .
- C. Ren, Thomas Grover Middle School