2024 AMC 8 Problems/Problem 15
Contents
- 1 Problem 15
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Answer Choices)
- 5 Video Solution (A Clever Explanation You’ll Get Instantly)
- 6 Video Solution 2 (easy to digest) by Power Solve
- 7 Video Solution 3 (2 minute solve, fast) by MegaMath
- 8 Video Solution 4 by SpreadTheMathLove
- 9 Video Solution by NiuniuMaths (Easy to understand!)
- 10 Video Solution by CosineMethod
- 11 Video Solution by Interstigation
- 12 Video Solution by Dr. David
- 13 See Also
Problem 15
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 can be would have to be , and it cannot be higher than that because 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 .
- Akhil Ravuri of John Adams Middle School
- Aryan Varshney of John Adams Middle School (minor edits... props to Akhil for the main/full answer :D)
~ cxsmi (minor formatting edits)
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
Solution 3 (Answer Choices)
Note that . Thus, we can check the answer choices and find through each of the answer choices, we find the 1107 works, so the answer is .
~andliu766
sus!
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/5ZIFnqymdDQ?si=rxqPhk-xiKjmbhNF&t=1683
~hsnacademy
Video Solution 2 (easy to digest) by Power Solve
Video Solution 3 (2 minute solve, fast) by MegaMath
https://www.youtube.com/watch?v=QvJ1b0TzCTc
Video Solution 4 by SpreadTheMathLove
https://www.youtube.com/watch?v=RRTxlduaDs8
Video Solution by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=V-xN8Njd_Lc
~NiuniuMaths
Video Solution by CosineMethod
https://www.youtube.com/watch?v=77UBBu1bKxk don't recommend but its quite clean, learn what you must- Orion 2010 minor edits by Fireball9746
Video Solution by Interstigation
https://youtu.be/ktzijuZtDas&t=1585
Video Solution by Dr. David
See Also
2024 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Problem 16 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.