Step-by-Step Factorization of 132 for Students
When we discuss the factor of 132, we truly mean all the positive and negative integers that can be divided into 132 with zero remainder. The result of taking the number 132 and dividing it by one of its factors would be another factor of 132.
Let's examine how to identify and list each of the 132 elements.
These numbers can be written as either individual factors or factor pairs. We present them both in this case. This is a specific number's mathematical decomposition.
How to Find Factors of 132?
First, we will divide 132 by the smallest prime number, 2. Then we will find the smallest prime that divides the quotient evenly. The process will continue until we get a quotient equal to 1. In this situation, the factors are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132, and the negatives -1, -2, -3, -4, -6, -11, -12, -22, -44, -66, -132 all of which can exactly divide 132. Factors are numbers that divide without leaving any remainder.
Prime Factors of 132
Sometimes, the prime factors of 132 and the prime factorisation of 132 are used interchangeably, which is strictly incorrect. Read this section carefully to understand the difference between the prime factors of 132 and the prime factorisation of 132.
The prime factorisation of 132 is 2 x 2 x 3 x 11. This is a simple and unique list of prime factors of 132, together with their multiplicities. 132 = 2 x 2 x 3 x 11.
The prime factors of 132 are the prime numbers that divide 132 exactly, with a remainder or modulo 0, in terms of Euclidean division.
2, 3, and 11 are the prime factors of 132.
The prime factorisation does not include 1, but it does include every copy of every prime factor. Now you know the decomposition of 132.
How to Find Prime Factors of a Number?
The prime factorisation can be done by two methods:
Division
Factor tree
The Division Method
The steps for determining the factors of any number are the same as those for determining the prime factors of that number.
To discover the number's smallest prime factor, divide it by the smallest prime number, 2, then by 3, then by 5, and so on.
Divide the quotient once again by the smallest prime.
Continue doing so until, after further division, the quotient equals 1.
Last but not least, show the number as the sum of all the prime components.
\[2 \times 2 \times 3 \times 11\] is the prime factorisation of 132. A composite number, 132. Composite numbers like 132 have at least two factorisations, in contrast to prime numbers, which only have one.
What is a Factor Tree?
A factor tree is a graphic representation of the possible factors of a number and its sub-factors. It is intended to make factorisation simpler. It is made by first determining the factors of a number and then the factors of the original number.
Once you get a large number of prime factors, which is the prime factorisation of the original number, the process recursively continues. Make sure to keep in mind the second component in a factor pair when building the tree. Now, see below the factor tree of 132.
Factor tree of 132
Finding the Factors of a Number
The following steps can be used to determine how many factors there are in a given number.
First, factor the provided number into primes, or describe it as the product of primes.
Write the prime factorisation in exponent form.
Increase each exponent by one.
Multiply each of the results. This product displays how many factors there are in the given integer.
Number of Factors of 132
We know that if a number can be expressed as a product of prime numbers, then by using the power of the prime factor, we can find the number of factors of that number.
Suppose \[N = X^{a}Y^{b}Z^{c}\], where X, Y, and Z are prime factors of N.
The number of factors of N is (a+1)(b+1)(c+1).
The prime factorization of 132 is \[132 = 2 x 2 x 3 x 11 = 2^{2} x 3 x 11\].
So the number of factors of 132 is \[(2+1)(1+1)(1+1) = 3 x 2 x 2 =12\].
The number of positive factors of 132 is 12.
Solved Examples
Example 1: Find the common factors of 132 and 151.
Sol: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132 are the factors of 132.
The factors 151 are 1 and 151.
As 151 is a prime number, the common factor of 132 and 151 is 1.
Example 2: What are the common factors of 132 and 143?
Sol: 132 factors are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132.
1, 11, 13, and 143 are 143 factors.
Thus, the common factors of 132 and 143 are 1 and 11.
Example 3: Find the 132 and 66 common factors.
Sol: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132 are the factors of 132.
1, 2, 3, 6, 11, 22, and 66 are the factors of 66.
Therefore, 1, 2, 3, 6, 11, 22, and 66 are the common factors of 132 and 66.
Conclusion
Despite being closely related, the terms "factors," "prime factors," and "prime factorisation of 132" have different meanings and cannot, strictly speaking, be used interchangeably.
A hundred and thirty-two has the following factors: 132, 66, 44, 33, 22, 12, 11, 6, 4, 3, 2, 1 and negatives of all these factors. The first three prime factors of 132 are 2, 3, and 11. Additionally, the prime factorisation of 132 is 2 x 2 x 3 x 11. A prime factor of 132 is not 1, so keep that in mind.
Key Features
132 is not a prime number but a composite number, and it can be written as the product of prime numbers:
The prime factorisation of the composite number 132:
132 = 2 × 2 × 3 × 11The factors are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132, and the negatives -1, -2, -3, -4, -6, -11, -12, -22, -44, -66, -132 all of these can exactly divide 132
List of Related Articles
Practice On Your Own
Q 1: Find the common factors of 142 and 132.
Ans: The common factors of 134 and 142 are 1 and 2.
Q 2: Find the common factors of 132 and 420.
Ans: The common factors of 132 and 420 are 1,2,3,4,6 and 12.
FAQs on Factors of 132 Explained with Examples
1. What are all the factors of 132?
The factors of 132 are all the whole numbers that can divide it exactly, leaving no remainder. The factors are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132. Each of these numbers can be multiplied by another whole number to result in 132.
2. What are the factor pairs of 132?
Factor pairs are sets of two numbers that, when multiplied together, equal 132. This method is a practical way to ensure you find all the factors. The positive factor pairs for 132 are:
- (1, 132)
- (2, 66)
- (3, 44)
- (4, 33)
- (6, 22)
- (11, 12)
3. How can you find the factors of 132 using prime factorization?
Prime factorization breaks a number down into its prime number components. You can create a factor tree to find this. The prime factorization of 132 is 2 × 2 × 3 × 11, or 2² × 3 × 11. To find all the other factors, you can create different combinations of these prime numbers (e.g., 2 × 2 = 4, 2 × 3 = 6, 2 × 11 = 22, and so on).
4. Why is 132 considered a composite number?
A number is called a composite number if it has more than two factors (meaning, factors other than just 1 and itself). Since 132 has twelve factors (1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132), it fits the definition of a composite number. A prime number, in contrast, has only two factors.
5. How are the factors of 132 different from its multiples?
This is a key distinction students must understand:
- Factors of 132 are numbers that divide it perfectly. Factors are always less than or equal to the number itself (e.g., 1, 2, 12, 66, 132).
- Multiples of 132 are numbers you get by multiplying 132 by another whole number. Multiples are always greater than or equal to the number itself (e.g., 132, 264, 396, 528...).
6. What is an easy way to check if a number is a factor of 132?
The easiest method is to use divisibility rules, which are mental math shortcuts. For 132, you can quickly check:
- Rule of 2: 132 is an even number, so 2 is a factor.
- Rule of 3: The sum of its digits (1 + 3 + 2 = 6) is divisible by 3, so 3 is a factor.
- Rule of 4: The last two digits (32) form a number divisible by 4, so 4 is a factor.
- Rule of 11: The alternating sum of its digits (2 - 3 + 1 = 0) is 0, so 11 is a factor.
7. In what real-life situation might you need to find the factors of 132?
Finding factors is essential for arranging items into equal groups. For example, if a teacher has 132 candies to distribute equally among students, the factors of 132 tell all the possible group sizes. The teacher could give 11 candies to 12 students, 4 candies to 33 students, or 2 candies to 66 students, ensuring no candies are left over.
8. What is the sum of all the factors of 132?
To find the sum of all the factors of 132, you add every factor together. The factors are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132. The sum is: 1 + 2 + 3 + 4 + 6 + 11 + 12 + 22 + 33 + 44 + 66 + 132 = 336. This type of calculation is a common problem in number theory exercises.
