Factors are numbers that can divide another number without leaving any remainder. For example, the factors 45 are the numbers that can divide by 45 without leaving any remainder. These numbers are 1, 3, 5, 9, 15, and 45. The number 1 and the number itself are always factors of any number. A number with more than two factors is called a composite number. Factors are important in many mathematical calculations and applications.
How to Find the Factors of a Number
There are two main ways to find the factors of a number:
1. Using the prime factorization of the number:
The prime factorization of a number is the expression of the number as the product of its prime factors. Prime factorization is a method used to express a given number as a product of its prime factors. Prime factors are the prime numbers that divide the original number without leaving a remainder. Finding the prime factorization of a number can be helpful in various mathematical operations, such as simplifying fractions or finding the greatest common divisor.
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Here are some steps for finding the prime factorization of 45:
Step 1: Divide 45 by the smallest prime number, 2.
When we divide 45 by 2, we get a quotient of 22 and a remainder of 1. This means that 2 is not a factor of 45, as it does not divide evenly.
Step 2: Move on to the following prime number, which is 3. We divide 45 by 3: 45 ÷ 3 = 15. Now we have a quotient of 15.
Step 3: Divide 15 by 3 again. Continuing the process, we divide 15 by 3 once more: 15 ÷ 3 = 5. Now we have a quotient of 5.
Step 4: Since 5 is a prime number, we cannot divide it further. At this point, we have obtained the prime factors of 45, which are 3 and 5. Both 3 and 5 are prime numbers and cannot be further divided.
Step 5: The prime factors from the division process are 3 and 5. We multiply the prime factors by 3 × 3 × 5, equal to 45. This is the prime factorization of 45.
Therefore, the prime factorization 45 is 3 × 3 × 5, or in exponent form, 3² × 5.
2. Using a factor tree:
Finding the factors of a number helps us understand all the numbers that divide evenly into it. One method to determine the factors is by using a factor tree. A factor tree is a graphical representation that breaks down a number into its prime factors, making it easier to identify all the factors.
Let’s use a factor tree to find the factors of 45:
Step 1: Write 45 at the top of the factor tree.
Step 2: Begin by finding two numbers that multiply to give 45. In this case, we can choose 5 and 9.
Step 3: Now, focus on the number 5 and see if it can be further factored. Since 5 is a prime number, we stop here.
Step 4: Move on to the number 9 and find its factors. We can choose 3 and 3, as they multiply to give 9.
Step 5: Since both 3’s are prime numbers, we stop here.
Step 6: We have reached the end of the factor tree. The numbers at the bottom represent the prime factors of 45.
Step 7: Read the prime factors from the bottom to the top of the tree. In this case, the prime factors are 5, 3, and 3.
The factor tree for 45 is shown below.
45
/ \
15 3
/ \ / \
5 9 1
The factor tree shows that factors 45 are 1, 3, 5, 9, 15, and 45.
Summary
In this article, we explain the factors of 45, which are numbers that divide evenly into it. The factors of 45 are 1, 3, 5, 9, 15, and 45. We explore two methods to find these factors: prime factorization and factor trees. Prime factorization breaks down 45 into prime factors (3 and 5), while factor trees visually represent them. Understanding factors helps in various calculations and applications.
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