A factor is a Latin word, which means “a doer” or “a maker” or “a performer.” In mathematics, a factor is a number, which divides the given number exactly. Thus, in another way, a factor is known as the divisor of a given number. A factor divides the given number with the remainder zero. 

The factor of any number can be found by using the multiplication method or division method. Factors are helpful in daily life activities, such as dividing quantities, arranging the objects in rows and columns and others. In this article, we will provide detailed information on factors. Continue reading to learn more about factors!

Define Factors

In mathematics, a factor is a number that divides the given number exactly, leaving a remainder zero. We can say that factor is known as the divisor of the given number. Factors may be numbers or algebraic expressions which divide the given numbers or algebraic expressions exactly. 

In number theory, each number has a minimum of two factors. The numbers with only two factors are called the prime numbers. The numbers with more than two factors are called composite numbers. 

The factors of a number can be found by using the multiplication method or division method.

When we multiply the two positive integers, we will get a positive integer as the product. Thus, the factors are multiplied to give the product.

In the given example, \(2 \times 4=8\).

The numbers \(2\) and \(4\) are known as the factors of the product \((8)\).

Factors can be used in various situations in our daily life, like dividing quantities, arranging things, such as placing sweets in the sweet box, arranging the objects or things in rows and columns, etc.

Factors and Multiples

At the primary level, the factors and multiples are the two main concepts that are studied together. Factors are the numbers that divide the given number exactly, whereas multiples are numbers that are multiplied by the other number to get specific numbers.

Example: \(5 \times 4=20\)

Here,
The numbers \(5\) and \(4\) divide the number \(20\) exactly. So, they are called factors.

The number \(20\) is known as multiple of each number, as it gives the product as \(20\) when we multiply \(5\) with \(4\) and vice-versa.

Common Factors

In mathematics, common factors are the factors that are common to two or more numbers. In another way, the common factors are the numbers, in which a set of two or more numbers divide the given numbers exactly, without leaving any remainder. Every number is divisible by one. So, one is the common factor of all the numbers.

Moreover, we know that two is the only number that divides all even numbers without leaving the remainder. So, the number two is the common factor of all the even numbers.

We know that the factors are the numbers that divide the given number exactly. 

To find the common factors of given two or more numbers, follow the below steps:

1. Write the factors of the given numbers separately.
2. Find the common factors present in those given numbers.

Example:
The common factors of \(12\) and \(16\) are \(1, 2\) and \(4\).

Properties of Factors

Properties of factors are listed below:

1. The number of factors of the given numbers is finite.
2. A factor of the given number is always less than or equals to the number given.
3. Every number has at least two factors, one \((1)\) and itself, except one \((1)\) and zero \((0)\).
4. The operations used to find the factors of the given numbers are division or multiplication.
5. One is the common factor of all numbers.
6. Prime numbers have only two factors, whereas composite numbers have more than two factors.

Finding Factors of Numbers by Division

The steps to be followed for finding the factors of the number by division as follows:

1. Consider the given number.
2. Consider all the numbers less than the given number and consider one and the number itself.
3. Divide the given number by each of the numbers taken.
4. Observe the remainder obtained.
5. If the remainder is zero, then the number is the factor of the given number.
6. Thus, take all the numbers, which divide the given number exactly, means leaving the remainder zero.

Example: The factors of \(6\) are found by division method as follows:

1. Taking all the numbers less than \(6\), i.e. \(1, 2, 3, 4, 5, 6\).
2. We are dividing \(6\) by each number and observing the remainder.
3. Only the numbers \(1, 2, 3, 6\) divides the number \(6\) exactly.
4. So, the numbers \(1, 2, 3, 6\) are the factors of \(6\).

Finding Factors of Numbers by Multiplication

The steps to be followed for finding the factors of the number by multiplication as follows:

1. Consider the given number.
2. Write the given number as the product of two numbers.
3. Write all such possible products of the given number.
4. Consider all the numbers involved in these products, which are known as factors of the given number.

Example:
The factors of the number \(24\) can be found by using multiplication as follows:

Expressing the \(24\) as the product of two numbers like \(1 \times 24=24,2 \times 12=24,3 \times 8=24,4 \times 6=24\) and vice-versa.

Thus, the numbers \(1,2,3,4,6,8,12,24\) are the factors of \(24\).

Finding Number of Factors

We can find the number of factors of the given number by following the below steps.

1. Find the prime factors of the given number by using the prime factorization method.
2. Express the given number as the product primes.
3. Write the prime factors in the exponent form.
4. Add one to each of the exponents.
5. Multiply all the numbers obtained after adding one to exponents.
6. The product provides the number of factors of the given number.

Example:
The number of factors of the number \(108\) found as follows:

1. Finding the prime factors of the number \(108\) as follows

2. Thus, \(108\) can be written as \(3 \times 3 \times 3 \times 2 \times 2=3^{3} \times 2^{2}\)
3. Adding one to each of the exponents and find the product of resultant numbers
\((3+1) \times(2+1)=4 \times 3=12\)
4. Thus, the number of factors of \(108\) is \(12\).

Importance of Factors

There are many uses associated with factors. The importance of factors in the field of Maths is listed below:

1. Dividing something into equal parts 
2. Dividing into rows and columns
3. Comparing prices
4. Exchanging money
5. Understanding time
6. Making calculations
7. During travel

Algebra of Factors

We can find factors for algebraic expressions as well. Some of the algebraic expressions can be expressed as the product of factors.

There are different types of methods to find the factors of algebraic expressions, such as:

1. Factoring expressions
2. Factoring polynomials
3. Factorization of quadratic expressions

Example:
The factors of the algebraic expression \(6xy\) can be found as follows:

\(6 x y=2 \times 3 \times x \times y\)

Thus, \(2,3, x, y\) are the factors of the given expression \(6 x y\).

Solved Examples on Factors

Q.1. Find the number of factors of \(1620\).
Ans: Given number is \(1620\).
The prime factors of the number \(1620\) can be found by using the prime factorization method as follows:

\(1620=2 \times 2 \times 5 \times 3 \times 3 \times 3 \times 3=2^{2} \times 3^{4} \times 5^{1}\)
We are adding one to each of the exponents and finding the product of new results formed.
\((2+1) \times(4+1) \times(1+1)=3 \times 5 \times 2=30\)
Therefore, the number of factors of \(1620\) is \(30\).

Q.2. Find the factors of \(12\).
Ans: The given number is \(12\).
Writing the \(12\) as the product of two numbers as follows:
\(1 \times 12=12\)
\(2 \times 6=12\)
\(3 \times 4=12\)
\(4 \times 3=12\)
\(6 \times 2=12\)
\(12 \times 1=12\)

All the numbers in these products are the factors of the given number.
Thus, the factors of \(12\) are \(1, 2, 3, 4, 6\) and \(12\).

Q.3. Find the common factor of the numbers \(2\) and \(5\).
Ans: The given numbers are the prime, and the factor of all prime numbers are one and themselves.
So, the factors of \(2\) are \(1,2\) and \(5\) are \(1,5\).
The common factor of the given numbers is \(1\).

Q.4. Find the factors of the number \(48\).
Ans: The given number is \(48\).
Write \(48\) as the product of two numbers in all possible ways.
\(48=1 \times 48\)
\(48=48 \times 1\)
\(48=2 \times 24\)
\(48=24 \times 2\)
\(48=3 \times 16\)
\(48=16 \times 3\)
\(48=4 \times 12\)
\(48=12 \times 4\)
\(48=6 \times 8\)
\(48=8 \times 6\)

All numbers which are in these products are the factors of the given number.
Therefore, the factors of the number \(48\) are \(1, 2, 3, 4, 6, 8, 12, 16, 24\) and \(48\).

Q.5. Find the factors of \(10\) by using division.
Ans: Take all numbers less than \(10-1,2,3,4,5,6,7,8,9\).
Divide \(10\) by each number and observe the remainder. If the remainder obtained is zero, then that number is a factor of the number.
Here, only the numbers \(1,2,5,10\) are dividing the number \(10\) exactly.

Therefore, the factors of \(10\) are \(1, 2, 5\) and \(10\).

Summary

In this article, we studied factors and their definition, which define the numbers that divide the given number exactly. A factor is defined as a number that divides a particular number exactly with a zero remainder. The factors are also useful for dividing something into equal parts and into rows and columns. It helps in comparing prices. Factors are also helpful in exchanging money.

There are several types of factors, like prime factors, linear factors, composite factors, common factors etc. Furthermore, the common factor is the factor that is common to the two or more numbers. It is also important to note that a factor of the given number is less than or equal to the number given.

FAQs on Factors

Q.1. What are the factors in Math?
Ans: In mathematics, a factor is a number that divides the given number exactly, with a remainder zero.

Q.2. What are all the factors of \(12\).
Ans: All factors of \(12\) are \(1, 2, 3, 4, 6\) and \(12\).

Q.3. How do you find the factors of a number?
Ans: The factors of the number are found by using the multiplication or division of the given numbers.

Q.4. What are common factors?
Ans: Common factors are the factors that are common to the two or more numbers.

Q.5. A prime number has how many factors?
Ans: The prime number has only two factors, one and the number itself.

Q.6. What are the different types of factors?
Ans: There are many types of factors, such as prime factors, linear factors, composite factors, common factors etc.

Practice on Le Chatelier’s Principle Questions

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