For how many integers, a, between 30 and 40 is it true that 5/a, 8/a, and 13/a are all in lowest terms?

Step 1: Understand the conditions for a fraction to be in lowest terms. A fraction $\frac{x}{a}$ is in lowest terms if the greatest common divisor (GCD) of $x$ and $a$ is 1, i.e., $\text{GCD}(x, a) = 1$. This means $x$ and $a$ share no common prime factors.

Step 2: List the integers $a$ between 30 and 40. The integers $a$ such that $30 < a < 40$ are: $31, 32, 33, 34, 35, 36, 37, 38, 39$

Step 3: Apply the condition for $\frac{5}{a}$ to be in lowest terms. For $\frac{5}{a}$ to be in lowest terms, $\text{GCD}(5, a) = 1$. This means $a$ cannot be a multiple of 5. From the list in Step 2, eliminate multiples of 5: $35$ is a multiple of 5. Remaining candidates: $31, 32, 33, 34, 36, 37, 38, 39$.

Step 4: Apply the condition for $\frac{8}{a}$ to be in lowest terms. For $\frac{8}{a}$ to be in lowest terms, $\text{GCD}(8, a) = 1$. The prime factors of 8 are only 2 ($8 = 2^3$). This means $a$ cannot be a multiple of 2, i.e., $a$ must be an odd number. From the remaining candidates in Step 3, eliminate even numbers: $32, 34, 36, 38$ are even. Remaining candidates: $31, 33, 37, 39$.

Step 5: Apply the condition for $\frac{13}{a}$ to be in lowest terms. For $\frac{13}{a}$ to be in lowest terms, $\text{GCD}(13, a) = 1$. This means $a$ cannot be a multiple of 13. From the remaining candidates in Step 4, eliminate multiples of 13: $39 = 3 \times 13$, so 39 is a multiple of 13. Remaining candidates: $31, 33, 37$.

Step 6: Count the number of integers that satisfy all conditions. The integers that satisfy all conditions are 31, 33, and 37. There are 3 such integers.

The correct option is (C) 3.

The number of integers is $\boxed{3}$.

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