Problem 15 - Discrete Mathematics with Applications

Question

Evaluate each sum, where \(d\) is a positive integer. $$\sum_{d | 6} d$$

Step-by-Step Solution

Answer

The sum of all the divisors of \(6\) is given by \(\sum_{d | 6} d = 1+2+3+6 = 12\).

Step 1: Find the Divisors of 6

First, we need to list all the divisors of \(6\). The divisors are the positive integers that can divide \(6\) without leaving a remainder. They are \(1\), \(2\), \(3\), and \(6\).

Step 2: Sum the Divisors

Next, we will add up all the divisors we found in step 1. The sum is given by: $$\sum_{d | 6} d = 1+2+3+6$$

Step 3: Calculate the Sum

Now, we simply evaluate the sum: $$1+2+3+6 = 12$$ So, the sum of all the divisors of \(6\) is \(12\).

Key Concepts

Divisors

In mathematics, divisors are numbers that divide another number completely, without leaving a remainder. This means if we have a number like 6, we look for all integers that can multiply with another whole number to result in 6. For instance:

• 1 is a divisor because 1 × 6 = 6.
• 2 is a divisor because 2 × 3 = 6.
• 3 is a divisor because 3 × 2 = 6.
• 6 is a divisor because 6 × 1 = 6.

Every positive integer is guaranteed to have divisors, as 1 and the number itself are always divisors. For the number 6, listing all divisors gives us 1, 2, 3, and 6. Recognizing divisors helps us understand and solve problems related to factorization and number properties.

Summation

Summation is the process of adding numbers together to get a total. It's a fundamental part of arithmetic and is denoted using the Greek letter sigma ( ∑ ). In the context of divisors, summation allows us to add all the divisors of a number together.

For the problem at hand, we work with ∑_{d | 6} d. This means we need to sum all the divisors of 6:

• 1 + 2 + 3 + 6 = 12.

This summation process aids in simplifying calculations, particularly when dealing with patterns or sequences. In many mathematical applications, such as finding averages or total values, summation is an essential technique.

Integer Arithmetic

Integer arithmetic refers to operations performed on whole numbers including addition, subtraction, multiplication, and division. Using integer arithmetic, one performs calculations with no fractions or decimals. It is crucial for basic math problems, especially those involving divisors and summation.

To find the sum of divisors:

• Add: Begin with the arithmetic operation of addition to combine numbers. Here, perform 1 + 2 + 3 + 6.
• Result: The sum equals 12, showcasing integer addition.

Since only whole numbers are involved, calculations are straightforward and robust. Understanding integer arithmetic gives the confidence to tackle higher-level math problems and proofs efficiently.