Difference between revisions of "2024 AMC 8 Problems/Problem 15"
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− | ==Problem== | + | ==Problem 15== |
Let the letters <math>F</math>,<math>L</math>,<math>Y</math>,<math>B</math>,<math>U</math>,<math>G</math> represent distinct digits. Suppose <math>\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}</math> is the greatest number that satisfies the equation | Let the letters <math>F</math>,<math>L</math>,<math>Y</math>,<math>B</math>,<math>U</math>,<math>G</math> represent distinct digits. Suppose <math>\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}</math> is the greatest number that satisfies the equation | ||
Line 9: | Line 9: | ||
==Solution 1== | ==Solution 1== | ||
− | The highest that <math>FLYFLY</math> can be would have to be <math>124124</math>, and it cannot be higher than that because then it would exceed the <math>6</math>-digit limit set on <math>BUGBUG</math>. | + | The highest that <math>FLYFLY</math> can be would have to be <math>124124</math>, and it cannot be higher than that, because then it would be <math>125125</math>, and <math>125125</math> multiplied by 8 is <math>1000000</math>, and then it would exceed the <math>6</math> - digit limit set on <math>BUGBUG</math>. |
− | So, if we start at <math>124124\cdot8</math>, we get <math>992992</math>, which would be wrong because both <math>B \& U</math> would be <math>9</math>, and the numbers cannot be repeated between different letters. | + | So, if we start at <math>124124\cdot8</math>, we get <math>992992</math>, which would be wrong because both <math>B \& U</math> would be <math>9</math>, and the numbers cannot be repeated between different letters. |
− | If we move on to the next | + | If we move on to the next highest, <math>123123</math>, and multiply by <math>8</math>, we get <math>984984</math>. All the digits are different, so <math>FLY+BUG</math> would be <math>123+984</math>, which is <math>1107</math>. So, the answer is <math>\boxed{\textbf{(C)}1107}</math>. |
- Akhil Ravuri of John Adams Middle School | - 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) | ~ cxsmi (minor formatting edits) | ||
+ | |||
+ | ~Alice of Evergreen Middle School | ||
==Solution 2== | ==Solution 2== | ||
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~andliu766 | ~andliu766 | ||
− | |||
− | |||
− | ==Video Solution | + | ==Video Solution by Math-X (Apply this simple strategy that works every time!!!)== |
− | https:// | + | https://youtu.be/BaE00H2SHQM?si=8CKIFksKvhOWABN3&t=3485 |
+ | |||
+ | ~MATH-x | ||
+ | |||
+ | ==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== | |
+ | https://youtu.be/TKBVYMv__Bg | ||
==Video Solution 3 (2 minute solve, fast) by MegaMath== | ==Video Solution 3 (2 minute solve, fast) by MegaMath== | ||
Line 66: | Line 77: | ||
~NiuniuMaths | ~NiuniuMaths | ||
− | == Video Solution by CosineMethod | + | == Video Solution by CosineMethod== |
https://www.youtube.com/watch?v=77UBBu1bKxk | 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== | ||
+ | https://youtu.be/kMkps16-xwE | ||
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/GJcAdyDYpQk | ||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2024|num-b=14|num-a=16}} | {{AMC8 box|year=2024|num-b=14|num-a=16}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | [[Category:Introductory Number Theory Problems]] |
Revision as of 06:46, 15 November 2024
Contents
[hide]- 1 Problem 15
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Answer Choices)
- 5 Video Solution by Math-X (Apply this simple strategy that works every time!!!)
- 6 Video Solution (A Clever Explanation You’ll Get Instantly)
- 7 Video Solution 2 (easy to digest) by Power Solve
- 8 Video Solution 3 (2 minute solve, fast) by MegaMath
- 9 Video Solution 4 by SpreadTheMathLove
- 10 Video Solution by NiuniuMaths (Easy to understand!)
- 11 Video Solution by CosineMethod
- 12 Video Solution by Interstigation
- 13 Video Solution by Dr. David
- 14 Video Solution by WhyMath
- 15 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 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 .
- 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)
~Alice of Evergreen 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
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
Video Solution by Math-X (Apply this simple strategy that works every time!!!)
https://youtu.be/BaE00H2SHQM?si=8CKIFksKvhOWABN3&t=3485
~MATH-x
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
Video Solution by WhyMath
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.