Problem 17 - Discrete Mathematics with Applications

Question

Evaluate each sum, where \(d\) is a positive integer. $$\sum_{d | 18}\left(\frac{1}{d}\right)$$

Step-by-Step Solution

Answer

The sum of the reciprocals of the divisors of 18 is \(\frac{13}{6}\).

Step 1: Find the divisors of 18

To find all positive divisors of 18, we can list the factors as pairs that multiply to give 18: 1 and 18, 2 and 9, 3 and 6. Combining these factors, we have the set of divisors: {1, 2, 3, 6, 9, 18}.

Step 2: Write down the sum of the reciprocals

Now we can write the sum of the reciprocals of the divisors of 18 as follows: \[ \sum_{d | 18}\left(\frac{1}{d}\right) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{18} \]

Step 3: Evaluate the sum

To evaluate the sum, we'll simply add the fractions: \[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{18} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{18} \] Now we find the common denominator, which is 18, and add the fractions: \[ \frac{18}{18} + \frac{9}{18} + \frac{6}{18} + \frac{3}{18} + \frac{2}{18} + \frac{1}{18} = \frac{18 + 9 + 6 + 3 + 2 + 1}{18} \] Adding the numerators: \[ \frac{39}{18} \] And finally, simplifying the fraction: \[ \frac{39}{18} = \frac{13}{6} \] So, the sum of the reciprocals of the divisors of 18 is \(\frac{13}{6}\).

Key Concepts

Divisors

In mathematics, divisors are numbers that divide another number completely without leaving a remainder. To find the divisors of a number, say 18, we need to search for all pairs of numbers that multiply to give 18. For example:

• 1 and 18
• 2 and 9
• 3 and 6

These pairs give us the complete set of divisors for 18: \( \{1, 2, 3, 6, 9, 18\} \). Divisors are useful for various mathematical techniques and are the foundation for finding other properties, like the reciprocals in this exercise.

Reciprocals

Reciprocals are the multiplicative inverse of a number. Essentially, the reciprocal of a number \( x \) is \( \frac{1}{x} \). For integers like divisors, it's simple: the reciprocal of 1 is \( \frac{1}{1} \), of 2 is \( \frac{1}{2} \), and so on.

• The set \( \{1, 2, 3, 6, 9, 18\} \) becomes \( \{\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{6}, \frac{1}{9}, \frac{1}{18}\} \) when considering reciprocals.

This concept is key in summing the reciprocals of divisors, as seen in this exercise.

Summation

Summation is the process of adding a sequence of numbers. In this exercise, we are summing the reciprocals of divisors of 18. The notation \( \sum_{d | 18}\left(\frac{1}{d}\right) \) denotes adding all the reciprocals where \( d \) is a divisor of 18.

• Start with the list: \( \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{18} \).
• Find a common denominator to combine the fractions.
• Add them to reach a final sum.

Summation is crucial in this context to find meaningful results from the reciprocals.

Fractions simplification

Simplifying fractions involves reducing them to their simplest form, where numerator and denominator have no common divisors except 1.

• For the sum \( \frac{39}{18} \), we divide both by their greatest common divisor (GCD), which is 3.
• This gives us \( \frac{39 ÷ 3}{18 ÷ 3} = \frac{13}{6} \).

By simplifying, we present the fraction in a more refined and understandable form, making further calculations or comparisons easier. This step completes our evaluation of the sum of reciprocals in the exercise.