Find Number, Sum, And Product Of Divisors Of 26 And 3600
Understanding divisors is a fundamental concept in number theory. In this comprehensive guide, we will explore how to find the number, sum, and product of divisors for given numbers. We will delve into the methods and formulas required to solve these types of problems, providing a step-by-step approach with examples. Specifically, we will focus on finding the divisors of 26 and 3600, illustrating the process with detailed explanations.
(i) Divisors of 26
Step 1: Prime Factorization
The first step in finding the divisors of a number is to perform prime factorization. Prime factorization is the process of expressing a number as a product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To find the prime factors of 26, we look for the prime numbers that divide 26 without leaving a remainder.
26 can be divided by 2, resulting in 13. Since 13 is also a prime number, the prime factorization of 26 is:
26 = 2 × 13
This prime factorization is crucial because it forms the basis for determining all the divisors of 26. The exponents of the prime factors (which are both 1 in this case, since 2 and 13 appear only once) will be used in subsequent calculations.
Step 2: Finding the Number of Divisors
To find the number of divisors, we use the prime factorization. If a number N can be expressed as
N = p₁ᵃ × p₂ᵇ × ... × pₙⁿ
where p₁, p₂, ..., pₙ are distinct prime factors and a, b, ..., n are their respective exponents, then the number of divisors d(N) is given by:
d(N) = (a + 1) × (b + 1) × ... × (n + 1)
For 26, the prime factorization is 2¹ × 13¹. Here, p₁ = 2, a = 1, p₂ = 13, and b = 1. Applying the formula:
d(26) = (1 + 1) × (1 + 1) = 2 × 2 = 4
Thus, 26 has 4 divisors. These divisors include 1 and the number itself (26), as well as the prime factors and their combinations. Understanding this method is essential for quickly determining the number of divisors for any integer.
Step 3: Listing the Divisors
Now that we know 26 has 4 divisors, we can list them by considering all possible combinations of its prime factors. The divisors are formed by taking each prime factor raised to powers from 0 up to its exponent in the prime factorization.
- 2⁰ × 13⁰ = 1 × 1 = 1
- 2¹ × 13⁰ = 2 × 1 = 2
- 2⁰ × 13¹ = 1 × 13 = 13
- 2¹ × 13¹ = 2 × 13 = 26
Therefore, the divisors of 26 are 1, 2, 13, and 26. Listing the divisors helps in understanding the composition of the number and is crucial for various mathematical problems, including finding the sum and product of divisors.
Step 4: Finding the Sum of Divisors
The sum of divisors, denoted as σ(N), can be calculated using the prime factorization. If N = p₁ᵃ × p₂ᵇ × ... × pₙⁿ, then the sum of divisors is given by:
σ(N) = (1 + p₁ + p₁² + ... + p₁ᵃ) × (1 + p₂ + p₂² + ... + p₂ᵇ) × ... × (1 + pₙ + pₙ² + ... + pₙⁿ)
Alternatively, the sum can be computed using the formula:
σ(N) = [(p₁ᵃ⁺¹ - 1) / (p₁ - 1)] × [(p₂ᵇ⁺¹ - 1) / (p₂ - 1)] × ... × [(pₙⁿ⁺¹ - 1) / (pₙ - 1)]
For 26, the prime factorization is 2¹ × 13¹. Using the formula:
σ(26) = (1 + 2) × (1 + 13) = 3 × 14 = 42
Alternatively, using the second formula:
σ(26) = [(2² - 1) / (2 - 1)] × [(13² - 1) / (13 - 1)] = (3 / 1) × (168 / 12) = 3 × 14 = 42
Thus, the sum of the divisors of 26 is 42. The sum of divisors is a significant property in number theory, used in various applications and theorems.
Step 5: Finding the Product of Divisors
The product of divisors, denoted as P(N), can also be determined using the number of divisors. If N has d divisors, then the product of divisors is given by:
P(N) = N^(d/2)
For 26, we found that the number of divisors d is 4. Therefore,
P(26) = 26^(4/2) = 26² = 676
So, the product of the divisors of 26 is 676. The product of divisors offers another perspective on the number's properties and its divisors' relationship.
(ii) Divisors of 3600
Step 1: Prime Factorization
Finding the divisors of 3600 involves the same steps as with 26, but with a larger number, the process is more detailed. We start with prime factorization.
3600 can be divided by 2 multiple times:
- 3600 ÷ 2 = 1800
- 1800 ÷ 2 = 900
- 900 ÷ 2 = 450
- 450 ÷ 2 = 225
So, 2 appears four times in the factorization. Now, 225 is not divisible by 2, but it is divisible by 3:
- 225 ÷ 3 = 75
- 75 ÷ 3 = 25
Thus, 3 appears twice. Finally, 25 is divisible by 5:
- 25 ÷ 5 = 5
So, 5 appears twice. Therefore, the prime factorization of 3600 is:
3600 = 2⁴ × 3² × 5²
This prime factorization is the foundation for the rest of the calculations, allowing us to determine the number of divisors, their sum, and their product.
Step 2: Finding the Number of Divisors
Using the formula for the number of divisors, where N = p₁ᵃ × p₂ᵇ × ... × pₙⁿ, and d(N) = (a + 1) × (b + 1) × ... × (n + 1), we apply it to 3600.
For 3600, the prime factorization is 2⁴ × 3² × 5². Here, p₁ = 2, a = 4, p₂ = 3, b = 2, p₃ = 5, and c = 2. Applying the formula:
d(3600) = (4 + 1) × (2 + 1) × (2 + 1) = 5 × 3 × 3 = 45
Therefore, 3600 has 45 divisors. Finding the number of divisors is crucial as it gives an idea of the complexity of the number's divisors and helps in various number-theoretic analyses.
Step 3: Listing the Divisors
Listing all 45 divisors of 3600 manually would be cumbersome. However, understanding the prime factorization allows us to know that the divisors will be formed by combinations of 2⁰ to 2⁴, 3⁰ to 3², and 5⁰ to 5². While a complete list is not practical here, the divisors include numbers like:
- 1 (2⁰ × 3⁰ × 5⁰)
- 2 (2¹ × 3⁰ × 5⁰)
- 3 (2⁰ × 3¹ × 5⁰)
- 4 (2² × 3⁰ × 5⁰)
- 5 (2⁰ × 3⁰ × 5¹)
- 6 (2¹ × 3¹ × 5⁰)
- 8 (2³ × 3⁰ × 5⁰)
- 9 (2⁰ × 3² × 5⁰)
- 10 (2¹ × 3⁰ × 5¹)
- ... and so on up to 3600
Listing a subset of divisors gives a sense of the range and distribution of the divisors.
Step 4: Finding the Sum of Divisors
The sum of divisors σ(N) for 3600 can be calculated using the formula:
σ(N) = [(p₁ᵃ⁺¹ - 1) / (p₁ - 1)] × [(p₂ᵇ⁺¹ - 1) / (p₂ - 1)] × ... × [(pₙⁿ⁺¹ - 1) / (pₙ - 1)]
For 3600 = 2⁴ × 3² × 5²:
σ(3600) = [(2⁵ - 1) / (2 - 1)] × [(3³ - 1) / (3 - 1)] × [(5³ - 1) / (5 - 1)]
σ(3600) = (31 / 1) × (26 / 2) × (124 / 4)
σ(3600) = 31 × 13 × 31 = 12493
Thus, the sum of the divisors of 3600 is 12493. Calculating the sum of divisors for larger numbers requires careful arithmetic and is an essential skill in number theory.
Step 5: Finding the Product of Divisors
The product of divisors P(N) for 3600 can be calculated using the formula P(N) = N^(d/2), where d is the number of divisors.
We found that 3600 has 45 divisors, so:
P(3600) = 3600^(45/2)
P(3600) = 3600^22.5
This result can be expressed as:
P(3600) = 3600²² × √3600
P(3600) = 3600²² × 60
Thus, the product of the divisors of 3600 is 3600²² multiplied by 60. The product of divisors grows rapidly with the number's size and the number of divisors, showcasing the multiplicative nature of divisors.
Conclusion
Finding the number, sum, and product of divisors involves understanding prime factorization and applying the appropriate formulas. For 26, we found 4 divisors with a sum of 42 and a product of 676. For 3600, we found 45 divisors, a sum of 12493, and a product of 3600²² × 60. This detailed guide provides a clear methodology for tackling such problems, reinforcing fundamental concepts in number theory and enhancing problem-solving skills.
