# 1960 AHSME Problems/Problem 16

## Problem

In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20,\ldots$. The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with: $\textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad \\ \textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad \\ \textbf{(E)}\ \text{four digits}$

## Solution

Since $25<69<125$, divide $69$ by $25$. The quotient is $2$ and the remainder is $19$, so rewrite the number as $$69 = 2 \cdot 25 + 19$$ Similarly, dividing $19$ by $5$ results in quotient of $3$ and remainder of $4$, so rewrite the number as $$69 = 2 \cdot 25 + 3 \cdot 5 + 4 \cdot 1$$. Thus, the number in base $5$ can be written as $234_5$, so the answer is $\boxed{\textbf{(C)}}$

## See Also

 1960 AHSC (Problems • Answer Key • Resources) Preceded byProblem 15 Followed byProblem 17 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 All AHSME Problems and Solutions
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