Problem 1378

1. We start with the given temperature conversion formula:
$$C = \frac{5}{9}(F - 32)$$
where $F$ is the temperature in Fahrenheit and $C$ is the temperature in Celsius.

2. We need to find integer values of $F$ such that $C$ is also an integer. For $C$ to be an integer, the expression $\frac{5}{9}(F - 32)$ must be an integer. This implies that $F - 32$ must be a multiple of 9.

3. We are given the range for $F$ as $45 \leq F \leq 70$. We need to find the values of $F$ within this range for which $F - 32$ is a multiple of 9.

4. Let $F - 32 = 9k$ for some integer $k$. Then:
$$F = 9k + 32$$

5. We substitute the range of $F$ into the equation:
$$45 \leq 9k + 32 \leq 70$$

6. Subtract 32 from all parts of the inequality:
$$13 \leq 9k \leq 38$$

7. Divide the entire inequality by 9:
$$\frac{13}{9} \leq k \leq \frac{38}{9}$$
$$1.444 \leq k \leq 4.222$$

8. Since $k$ must be an integer, the possible values of $k$ are 2, 3, and 4.

9. We now find the corresponding values of $F$:
$$\begin{align*} k = 2 & : F = 9(2) + 32 = 18 + 32 = 50 \\ k = 3 & : F = 9(3) + 32 = 27 + 32 = 59 \\ k = 4 & : F = 9(4) + 32 = 36 + 32 = 68 \\ \end{align*}$$

10. Therefore, the temperature is an integer in both Fahrenheit and Celsius at three different times: when $F = 50^\circ \text{F}$, $F = 59^\circ \text{F}$, and $F = 68^\circ \text{F}$.

The final answer is 3