HMMT November, 2023

Number theory Difficulty 4.4 AIME Find the answer United States

Problem:

The number $5.6$ may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} \cdot \underline{d}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

We want $\overline{a b} \times \overline{c d} = 560 = 2^{4} \times 5 \times 7$ . To avoid a zero digit, we need to group the $5$ with the $7$ to get $3.5$ and $1.6$ , and our answer is $3.5 + 1.6 = 5.1$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.