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Difference between revisions of "2024 AMC 8 Problems/Problem 15"

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We can now use our equation to test each answer choice.
 
We can now use our equation to test each answer choice.
  
−
We have that <math>124124 \times 8 = 992992</math>, so we can find the sum:
+
We have that <math>123123 \times 8 = 984984</math>, so we can find the sum:
  
−
<math>\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G} = 124 + 992= 1116</math>.
+
<math>\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G} = 123 + 984= 1107</math>.
  
−
So, the correct answer is <math>\textbf{(D)}\ 1116</math>.
+
So, the correct answer is <math>\textbf{(C)}\ 1107</math>.
  
 
==Video Solution by Math-X (First fully understand the problem!!!)==
 
==Video Solution by Math-X (First fully understand the problem!!!)==

Revision as of 13:46, 25 January 2024

Problem

Let the letters $F$,$L$,$Y$,$B$,$U$,$G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation

\[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\]

What is the value of $\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}$?

$\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125$

Solution

Notice that $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y} = 1000(\underline{F}~\underline{L}~\underline{Y}) + \underline{F}~\underline{L}~\underline{Y}$.

Likewise, $\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G} = 1000(\underline{B}~\underline{U}~\underline{G}) + \underline{B}~\underline{U}~\underline{G}$.

Therefore, we have the following equation:

$8 \times 1001(\underline{F}~\underline{L}~\underline{Y}) = 1001(\underline{B}~\underline{U}~\underline{G})$.

Simplifying the equation gives

$8(\underline{F}~\underline{L}~\underline{Y}) = (\underline{B}~\underline{U}~\underline{G})$.

We can now use our equation to test each answer choice.

We have that $123123 \times 8 = 984984$, so we can find the sum:

$\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G} = 123 + 984= 1107$.

So, the correct answer is $\textbf{(C)}\ 1107$.

Video Solution by Math-X (First fully understand the problem!!!)

https://www.youtube.com/watch?v=JK4HWnqw-t0

~Math-X

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