# 1987 AHSME Problems/Problem 5

## Problem

A student recorded the exact percentage frequency distribution for a set of measurements, as shown below. However, the student neglected to indicate $N$, the total number of measurements. What is the smallest possible value of $N$? $$\begin{tabular}{c c}\text{measured value}&\text{percent frequency}\\ \hline 0 & 12.5\\ 1 & 0\\ 2 & 50\\ 3 & 25\\ 4 & 12.5\\ \hline\ & 100\\ \end{tabular}$$ $\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 50$

## Solution

Note that $12.5\% = \frac{1}{8}$, $25\% = \frac{1}{4}$, and $50\% = \frac{1}{2}$. Thus, since the frequencies must be integers, $N$ must be divisible by $2$, $4$, and $8$ (so that $\frac{N}{8}$ etc. are integers), or in other words, $N$ is divisible by $8$. Thus the smallest possible value of $N$ is the smallest positive multiple of $8$, which is $8$ itself, or $\boxed{\text{B}}$.

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