Problem 37
Question
Let \(n \in \mathbb{N}\) . A positive integer \(d\) is a proper factor of \(n\) if \(d\) is a factor of \(n\) and \(d < n .\) For example, the proper factors of 12 are \(1,2,3,4,\) and 6 . Let \(\sigma : N \rightarrow N\) defined by \(\sigma(n)=\) sum of the proper factors of \(n .\) (\sigma is the lowercase Greek letter, sigma.) Compute \(\sigma(n)\) for each value of \(n,\) where \(p\) and \(q\) are distinct primes, IA positive integer \(n\) such that \(\sigma(n)=n\) is a perfect number. $$ 6 $$
Step-by-Step Solution
Verified Answer
The proper factors of \(n=6\) are 1, 2, and 3. We can compute the sum of these factors as \(1 + 2 + 3 = 6\). Therefore, \(\sigma(6) = 6\).
1Step 1: List all positive factors of n
Find all numbers that can divide 6. Here are the factors for \(n=6\): 1, 2, 3, and 6.
2Step 2: Identify the proper factors of n
The proper factors are the factors of 6 that are less than 6. These are: 1, 2, and 3.
3Step 3: Compute the sum of the proper factors of n
Add up the proper factors: \(1+2+3\). This equals 6.
The function \(\sigma(n)\) for \(n=6\) is equal to the sum of its proper factors, which is 6.
Key Concepts
Sigma FunctionPerfect NumberPrime Number
Sigma Function
The sigma function, denoted as \( \sigma(n) \), is a significant concept in the realm of discrete mathematics. It is defined as the sum of the proper factors of a positive integer \( n \). Proper factors are those divisors of \( n \), excluding the number itself, that can be divided into it without leaving a remainder. For example, for \( n = 10 \), its proper factors are 1, 2, and 5. Applying the sigma function, we get \( \sigma(10) = 1 + 2 + 5 = 8 \).
Understanding the sigma function is crucial because it helps us explore interesting characteristics of numbers, such as identifying perfect numbers and understanding number divisibility. It is an invaluable tool for researchers and enthusiasts who delve into number theory.
Understanding the sigma function is crucial because it helps us explore interesting characteristics of numbers, such as identifying perfect numbers and understanding number divisibility. It is an invaluable tool for researchers and enthusiasts who delve into number theory.
Perfect Number
A perfect number is a positive integer that is equal to the sum of its proper factors. This concept is deeply entwined with the sigma function as it is through this function we can identify whether a number is perfect. To demonstrate, consider the exercise involving \( n = 6 \). Its proper factors are 1, 2, and 3. When we calculate the sum using the sigma function (\( \sigma(6) \)), we find that it equals 6, the number itself, thus classifying it as a perfect number.
Perfect numbers have been studied for centuries, with early references to them appearing in ancient Greek mathematics. They are relatively rare and have a harmonic quality often described as mathematical beauty. The exploration of perfect numbers leads to exciting questions about their distribution, rarity, and properties.
Perfect numbers have been studied for centuries, with early references to them appearing in ancient Greek mathematics. They are relatively rare and have a harmonic quality often described as mathematical beauty. The exploration of perfect numbers leads to exciting questions about their distribution, rarity, and properties.
Prime Number
A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In other words, a prime number \( p \) has exactly two distinct factors: 1 and \( p \). This attribute is crucial when working with proper factors, as a prime number's only proper factor is 1, making its sigma value \( \sigma(p) = 1 \).
Prime numbers are the building blocks of all integers because any number can be expressed as a unique product of primes, known as its prime factorization. This fundamental property is a cornerstone of arithmetic and has implications in various fields like cryptography and computer science. Understanding primes is essential in the context of discrete mathematics, especially when dealing with numbers' divisibility and structure.
Prime numbers are the building blocks of all integers because any number can be expressed as a unique product of primes, known as its prime factorization. This fundamental property is a cornerstone of arithmetic and has implications in various fields like cryptography and computer science. Understanding primes is essential in the context of discrete mathematics, especially when dealing with numbers' divisibility and structure.
Other exercises in this chapter
Problem 37
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