Difference between revisions of "2024 AMC 8 Problems/Problem 15"
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<math>\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125</math> | <math>\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125</math> | ||
+ | |||
+ | ==2 minute solve (fast) by MegaMath== | ||
+ | https://www.youtube.com/watch?v=QvJ1b0TzCTc | ||
==Solution 1== | ==Solution 1== |
Revision as of 11:52, 26 January 2024
Contents
[hide]Problem
Let the letters ,,,,, represent distinct digits. Suppose is the greatest number that satisfies the equation
What is the value of ?
2 minute solve (fast) by MegaMath
https://www.youtube.com/watch?v=QvJ1b0TzCTc
Solution 1
The highest that can be would have to be , and it cannot exceed that because it would exceed the -digit limit set on .
So, if we start at , we get , which would be wrong because the numbers cannot be repeated.
If we move on to and multiply by , we get , all the digits are different, so would be , which is . So, the answer is .
-Akhil Ravuri, John Adams Middle School
~ 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, Thomas Grover Middle School
Video Solution by Math-X (First fully understand the problem!!!)
https://www.youtube.com/watch?v=JK4HWnqw-t0
~Math-X