Question

Let the letters $F,$ $L,$ $Y,$ $B,$ $U,$ $G$ represent distinct digits. Suppose $\text{FLYFLY}$ is the greatest number that satisfies the equation $8 \cdot \text{FLYFLY} = \text{BUGBUG}.$ What is the value of $\text{FLY} + \text{BUG}?$

$1089$

$1098$

$1107$

$1116$

$1125$

Answer: C

Concepts:cryptarithm place value optimization

Difficulty rating: 1540

Video solution:

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Written solution:

Firstly, note that $\text{FLYFLY} = 1001(\text{FLY})$ and, similarly, $\text{BUGBUG}=1001(\text{BUG})$ so the equation can be simplified to $8 \cdot \text{FLY} = \text{BUG}.$

Because $\text{BUG}$ has three digits, $\text{FLY}\le124.$ The largest possible three-digit number at most $124$ with distinct digits is $124,$ but $8\cdot124=992$ repeats a digit and also reuses the digit $2.$

The next candidate is $123,$ and $8\cdot123=984.$ The six digits $1,2,3,9,8,4$ are all distinct, so $123$ is the greatest possible value of $\text{FLY}.$

Hence, $\begin{gathered} \text{FLY} + \text{BUG} = 123 + 984 \\ = 1107. \end{gathered}$

Thus, C is the correct answer.

2024 AMC 8 Problem 15

Answer: C

Difficulty rating: 1540

Video solution:

Written solution:

Problem 15 in Other Years