Question

Consider the set of all positive rational numbers that are less than 1 and that have denominators as 30 in their lowest terms. Their sum is equal to

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

Number Theory Tip: For any denominator $n$, the sum of all proper fractions $\frac{k}{n}$ in their lowest terms is always exactly $\frac{\phi(n)}{2}$, where $\phi(n)$ is the count of numbers coprime to $n$. Since there are 8 coprime numbers here, the sum is $8/2 = 4$.

Answer 1

The Correct Option is D

Concept:
For a rational number $\frac{k}{30}$ to be in its "lowest terms," the numerator $k$ and the denominator $30$ must be coprime, meaning their greatest common divisor is 1 ($\gcd(k, 30) = 1$). To find the valid numbers, we must eliminate any $k$ that shares prime factors with 30.

Step 1: Identify the prime factors of the denominator.
The denominator is 30. We break it down into its prime factorization: $$30 = 2 \times 3 \times 5$$ This means valid numerators $k$ cannot be multiples of 2, 3, or 5.

Step 2: List the valid numerators.
We need to find all integers $k$ such that $1 \le k < 30$ and $k$ is not divisible by 2, 3, or 5. Checking numbers up to 30, we eliminate evens, multiples of 3, and numbers ending in 5 or 0. The remaining valid numerators are: $$k \in \{1, 7, 11, 13, 17, 19, 23, 29\}$$

Step 3: Set up the sum of these fractions.
The problem asks for the sum of all these valid rational numbers: $$\text{Sum} = \frac{1}{30} + \frac{7}{30} + \frac{11}{30} + \frac{13}{30} + \frac{17}{30} + \frac{19}{30} + \frac{23}{30} + \frac{29}{30}$$

Step 4: Group the numerators to simplify addition.
Notice that the numerators form symmetric pairs that perfectly add up to 30: $1 + 29 = 30$ $7 + 23 = 30$ $11 + 19 = 30$ $13 + 17 = 30$

Step 5: Calculate the final sum.
Substitute the sum of these pairs back into the fractional expression: $$\text{Sum} = \frac{30 + 30 + 30 + 30}{30}$$ $$\text{Sum} = \frac{120}{30}$$ $$\text{Sum} = 4$$ Hence the correct answer is (D) 4.