Problem 37 - Fundamentals of Mathematics

Question

Find all the factors of each of the numbers. 105

Step-by-Step Solution

Answer

The factors of 105 are 1, 3, 5, 7, 15, 21, 35, 105.

Step 1: Determine the factors of 105

To determine the factors of 105, we start by dividing 105 by the smallest prime number, which is 2. Since 105 is odd, it is not divisible by 2. We then try the next prime number, which is 3. So, we divide 105 by 3: \[ 105 \div 3 = 35 \]Since 35 is still greater than 1, our next task is to find the factors of 35 as well.

Step 2: Determine the factors of 35

Continuing from the previous step with 35, we look for its smallest divisor. It is not divisible by 2 or 3. We try the next prime number, which is 5. So, we divide 35 by 5:\[ 35 \div 5 = 7 \]Since 7 is also a prime number, the factorization stops here. We now have all prime factors: 3, 5, and 7.

Step 3: List the factors of 105

Now that we have determined all the prime factors of 105, we can find all the factors by multiplying these in different combinations:- Using no multiplication: 1 (always a factor).- Using one factor: 3, 5, 7.- Using two factors: \(3 \times 5 = 15\), \(3 \times 7 = 21\), \(5 \times 7 = 35\).- Using three factors: \(3 \times 5 \times 7 = 105\).Thus, the complete list of factors of 105 is: 1, 3, 5, 7, 15, 21, 35, 105.

Key Concepts

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. This means they cannot be formed by multiplying two smaller natural numbers. To find the prime factorization of a number, follow these steps:

• Start with the smallest prime number, usually 2. Check if the number is divisible by it.
• If it isn't divisible, move to the next smallest prime numbers like 3, 5, 7, etc.
• Divide the original number by its prime factors until you reach 1 or a prime number itself.

Let's consider the example of 105. Since 105 is odd, it is not divisible by 2. By testing 3 and 5, we find that 105 can be expressed as a product of the prime numbers 3, 5, and 7. Thus, the prime factorization of 105 is: \[ 105 = 3 imes 5 imes 7 \] By mastering prime factorization, you gain a tool that is useful for various mathematical applications, including finding the greatest common divisor and understanding number properties.

Multiples

Understanding multiples is essential when working with factors. A multiple of a number is the result of multiplying that number by an integer. For example, multiples of 3 include 3, 6, 9, 12, and so on. This process involves simple multiplication, where:

• The number itself is the first multiple.
• Next multiples are found by multiplying the number by 2, 3, 4, etc.

Considering our number 105, some of its multiples would be 105, 210, 315, etc., which are calculated by multiplying 105 with whole numbers (1, 2, 3,...). When dealing with factors, remember that factors of a number divide it evenly, while its multiples are found by multiplying. Multiples expand outward while factors narrow down a number's divisors.

Divisibility

Divisibility refers to whether a number can be evenly divided by another without leaving a remainder. It's a key concept to understand when searching for factors, as it determines which numbers can perfectly divide your target number. Key rules for checking divisibility include:

• A number is divisible by 2 if it is even.
• It's divisible by 3 if the sum of its digits is a multiple of 3.
• A number is divisible by 5 if it ends in 0 or 5.

In the exercise, to determine that 105 is divisible by 3, we check the sum of its digits (1 + 0 + 5 = 6), which is a multiple of 3. Understanding these rules helps simplify the factorization process by eliminating non-divisors quickly.

Mathematical Operations

The process of finding factors involves various mathematical operations, but primarily it is about division and multiplication. Here's how these operations come into play:

• Division: When looking for factors, you repeatedly divide the main number by other numbers to see if they are factors.
• Multiplication: Once the prime factors are identified, you multiply them in different combinations to list all possible factors.

For example, in the solution for finding factors of 105, we use division to check which numbers can divide 105 without a remainder. Then, we use multiplication to find which combinations of the prime factors (3, 5, and 7) will give back the original number when multiplied, as well as other smaller factors like 1, 15, 21, 35, and 105. Engaging with these operations enhances computational skills and deepens understanding of the relationship between numbers.