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

Revision as of 17:28, 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 1

The highest that $FLYFLY$ can be would have to be $124124$ , and it cannot exceed that because it would exceed the $6$ -digit limit set on $BUGBUG$ .

So, if we start at $124124\cdot8$ , 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, the answer is $\boxed{\textbf{(C)}1107}$ .

-Akhil Ravuri, John Adams Middle School

~ cxsmi (minor formatting edits)

Solution 2

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 $\textbf{(C)}\ 1107$ .

- C. Ren, Thomas Grover Middle School

Video Solution 1 (easy to digest) by Power Solve

https://youtu.be/TKBVYMv__Bg

Video Solution by Math-X

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

@@ Line 43: / Line 43: @@
 - C. Ren, Thomas Grover Middle School
-==Video Solution 1 (clear + straightforward) by Power Solve==
+==Video Solution 1 (easy to digest) by Power Solve==
 https://youtu.be/TKBVYMv__Bg
 ==Video Solution by Math-X==
 https://www.youtube.com/watch?v=JK4HWnqw-t0

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