If `a_1,a_2,a_3,.......a_n`, are 'n', distinct odd natural numbers, not divisible by any prime number greater than 5, then `1/a_1+1/a_2+1/a_3+......+1/a_n` is less than `(a) 15/8 (b) 17/8 (c)19/8 (d) 21/8`

Options

A. 1
B. `15//8`
C. `1//2`
D. `3//4`

Explanation

To solve the problem, we need to analyze the sum of the reciprocals of distinct odd natural numbers that are not divisible by any prime number greater than 5. The odd natural numbers that fit this criterion are 1, 3, 5, 15, etc. However, since we are only considering distinct odd natural numbers, we will focus on the prime factors 3 and 5.

### Step-by-step Solution:

1. **Identify the odd natural numbers**: The odd natural numbers that are not divisible by any prime number greater than 5 are formed from the prime factors 3 and 5. The possible odd numbers are:
- \(1\) (which is not considered since it does not contribute to the sum)
- \(3\)
- \(5\)
- \(15\) (since \(15 = 3^1 \times 5^1\))
- Higher combinations like \(3^2 = 9\) and \(5^2 = 25\) are not considered since they are not odd or exceed the limit of being distinct.

2. **Set up the sum of reciprocals**: We need to find the sum \(S = \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n}\) where \(a_i\) are the distinct odd natural numbers identified above.

For our case, we can take:
- \(a_1 = 3\)
- \(a_2 = 5\)
- \(a_3 = 15\)

Thus, the sum becomes:
\[ S = \frac{1}{3} + \frac{1}{5} + \frac{1}{15} \]

3. **Calculate the sum**:
- First, find a common denominator for the fractions. The least common multiple (LCM) of \(3\), \(5\), and \(15\) is \(15\).
- Rewrite each fraction with the common denominator:
\[ S = \frac{5}{15} + \frac{3}{15} + \frac{1}{15} = \frac{5 + 3 + 1}{15} = \frac{9}{15} \]
- Simplifying gives:
\[ S = \frac{3}{5} \]

4. **Compare with the options**: Now we need to check if this sum \(S\) is less than any of the given options:
- \( \frac{15}{8} = 1.875\)
- \( \frac{17}{8} = 2.125\)
- \( \frac{19}{8} = 2.375\)
- \( \frac{21}{8} = 2.625\)

Since \( \frac{3}{5} = 0.6\), it is indeed less than all of the options provided.

5. **Conclusion**: Therefore, the correct answer is \( \frac{15}{8} \) (option a).