2008 Mock ARML 1 Problems/Problem 7

Problem

Compute the number of $3$ -digit base- $5$ positive integer multiples of $7$ that are also divisible by $7$ when read in base $10$ instead of base $5$ .

Solution

Let the number be $\overline{abc}$ . Then $7 | 25a + 5b + c, 100a + 10b + c$ , and so it must divide their difference, so $7 | 75a + 5b \Longrightarrow 7|-2(a+b)$ , from which it follows that $7|a+b$ . However, as $a,b < 5$ , we have $\{a,b\} = \{3,4\}$ , leading to $\boxed{2}$ solutions: $343, 434$ .

2008 Mock ARML 1 (Problems, Source)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8

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2008 Mock ARML 1 Problems/Problem 7

Problem

Solution

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