Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Find the answer

6. As shown in Figure 1, each number in a box is determined by the product of the two numbers in the adjacent boxes of the upper layer (for example, $30=6 \times 5$ ). Then the number that should be filled in the unfilled box in the first row is ( ).
(A) $2 \quad$ (B) 3
$\begin{array}{lll}(\mathrm{C}) 4 & (\mathrm{D}) 5 & (\mathrm{E}) 6\end{array}$

This was a multiple-choice question, but the options didn't survive into the source we have. The answer given is C, and the solution below works it through.

Solution

6. C.

Let the unfilled number in the first row be $x$ , and the unfilled number in the second row be $y$ . According to the problem, we have
$\left\{\begin{array} { l } { 6 0 0 = 3 0 y , } \\ { y = 5 x } \end{array} \Rightarrow \left\{\begin{array}{l} x=4, \\ y=20 . \end{array}\right.\right.$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.