Prime Factorization: Factor Tree Method & Division Method

“Prime Factorization” is finding which prime numbers multiply together to make the original number. This process of finding the prime factors of a given number is known as prime factorization. To understand it better, first, we need to know about the meaning of prime numbers and the factors.

A Prime Number is a number of which has only two factors: \(1\) and itself. For example \(2, 3, 5, 7, 13,\) etc. It is a whole number greater than 1 that can not be made by multiplying other whole numbers.

Factors are the numbers that divide another number completely with no remainder. They are the numbers you multiply together to get another number.

In simple terms, prime factorization is a method of finding which prime numbers multiply together to make the original number. With this article, one can learn how to find the prime factorization of a given number. Read on to find out more.

What is Prime Factorization?

The method of writing a number as the product of prime numbers is known as prime factorization. To find the prime factors of a given number, the number should be divided by prime numbers until the remainder is equal to \(1.\)

What are Prime numbers?

The natural numbers which are greater than \(1\) and have only two factors, \(1\) and the number itself are called prime numbers.
The term “greater than \(1\) is needed in the definition of a prime number to exclude \(1.\)
Examples of prime numbers are \(2,\;3,\;5,\;7,\;11,\;13,\;\) etc.

What are Factors?

The natural numbers that we multiply together to get the product as another number are called factors of that number. This means a factor divides a number evenly without leaving any remainder.
Example: \(2 \times 8 = 16.\) Here \(2\) and \(8\) are the factors of \(16.\)

What is Factorization?

The process of splitting a number into a product of factors, which when multiplied together gives the original number is known as factorization.
Example: \(48 = 6 \times 8,\) here \(6\) and \(8\) are the factors of \(48.\)
\(16 = 4 \times 4,\) here \(4\) is the factor of \(16.\)
The process of splitting \(48\) to \(6\;\& \;8\) is called factorisation.

What are the Prime Factors of a Number?

The prime factor is the factor of a given number which is a prime number. The numbers we multiply together to get another number are called factors. In other words, the prime factors are the prime numbers when multiplied together gives the actual number.

For Example: The prime factors of \(21\) are \(3\) and \(7,\) because \(3 \times 7 = 21,\) and \(3\) and \(7\) are Prime numbers.

What are the types of Prime Factorization?

The two commonly used methods of the prime factorization are:
1. Factor tree method.
2. Division method.

Prime Factorization using the Factor Tree Method?

We know that prime factorization means finding the unique set of prime numbers that when multiplied gives the composite number.
To do a prime factorization, one of the methods is a factor tree method. This method helps to split the composite number into its prime factors in a step by step manner.

To perform a factor tree method, first, we need to consider the given composite number as the root of the tree. The trees have branches, so write the pair of factors as the branches of a tree. If we find any composite number in the branch, split that composite number again as a pair of factors. Repeat this process until we find both the factors of a given composite number are prime numbers.

Example: Let us consider a composite number \(18.\) We know that the factors of \(18.\) are \(2 \times 9.\) Here, \(2\) is a prime number and \(9\) is a composite number. So, we need to find the factors of \(9.\) The factors of \(9\) are \(3 \times 3.\) Here, \(3\) is a prime number. So, there is no composite number left in the above process.

Also, the factors of \(18\) are \(3 \times 6.\) Here, \(3\) is a prime number and \(6\) is a composite number. So, we need to find the factors of \(6.\) The factors of \(6\) are \(3 \times 2.\) Here, \(3\) and \(2\) both are prime numbers. So, there is no composite number left in the above process.

Let us show the above process in a tree factor method.

Now, we have found all the prime factors of \(18,\) So the prime factors of \(18\) are \(2 \times 3 \times 3.\) It is correct but not yet done. To understand in a better manner, we represent the repeated prime numbers in an exponential form. We call it exponential notation.
\(\Rightarrow 2 \times 3 \times 3 = 2 \times {3^2}\)
So, that is the standard way of listing prime factors using the factor tree method.

What is the meaning of Prime Factorization using the Division Method?

The division method is the fastest way to find the prime factors of any given number. We know that the first primes are \(2,\;3,\;5,\;7,\;11,\;13,\;17.\) When we do a prime factorization using division, we can only use prime numbers as our divisors.
To perform the division method, first, we need to divide the given number by any of the smallest prime numbers, again divide the quotient by the smallest prime number. Continue this process until the quotient becomes \(1.\) Then, multiply all the prime factors.
Example: Let us find the prime factors of \(252.\)

We know that \(252\) is an even number that is divisible by \(2\) So first let us find \(252 \div 2\) The value of \(252 \div 2 = 126.\) Now, \(126\) is also an even number that is divisible by \(2\). So, \(126 \div 2 = 63.\) Now, \(63\) is an odd number and \(63\) is divisible by \(3.\). So, \(63 \div 3 = 21\) and also \(21\) is divisible by \(3.\). So, \(21 \div 3 = 7\) and \(7 \div 7 = 1.\). So the quotient is \(1.\).

The above explanation can be done step by step by the below process.

Therefore, the factors of \(252\) are \(2522 \times 2 \times 3 \times 3 \times 7.\)
The exponential form of the prime factors of \(252 = {2^2} \times {3^2} \times 7.\)

Solved Examples – Prime Factorization

Q.1. Find the prime factors of \(7429\) using prime factorization.
Ans: We can find the prime factorization of \(7429\) in both ways.
Division method: The first prime number which divides \(7429\) is \(17.\) The quotient of \(7429 \div 17\) is \(437.\) The prime number which divides \(437\) is \(19\) the quotient of \(437 \div 19\) is \(23.\) Now, \(23\) is a prime number. Therefore, \(23 \div 23 = 1.\)

Factor tree method: Below is the factor tree method to find the prime factors of \(7429.\)

Therefore, the prime factors of \(7429\) are \(17 \times 19 \times 23.\)

Q.2. Find the prime factors of \(60\) using prime factorization?
Ans: We can find the prime factorization of \(60\) in both ways.
The first prime number which divides \(60\) is \(2.\) So, \(60 \div 2 = 30.\) Since \(30\) is an even number, it is divisible by \(2.\) So, \(30 \div 2 = 15.\) Now, \(15\) is divisible by the second prime number \(3.\) So, \(15 \div 3 = 5.\) Here, \(5\) is a prime number, so \(5 \div 5 = 1\)

Factor tree method: Below is the factor tree method to find the prime factors of \(60.\)

So, the prime factors of \(60\) are \(2 \times 2 \times 3 \times 3 \times 5.\) Its exponential form is \({2^2} \times {3^2} \times 5.\)

Q.3. Find the prime factors of \(5005\) using prime factorization.
Ans: We can find the prime factorization of \(5005\) in both ways.
Since \(5005\) ends with \(5,\) it is known that \(5005\) will be divisible by the prime number \(5.\) So, \(5005 \div 5 = 1001.\) 1001 is divisible by \(7.\) So, \(1001 \div 7 = 143.\) Now, \(143\) is divisible by \(11.\) So, \(143 \div 11 = 13\) and \(13\) is a prime number. So, \(13 \div 13 = 1.\)
Below is the division method to find the prime factors of \(5005.\)

Below is the factor tree method to find the prime factors of \(5005.\)

Q.4. Find the prime factors of \(999\) using the division method.
Ans: We know that the first prime number which divides \(999\) is \(3.\) So, \(999 \div 3 = 333.\) Also, \(333\) is divided by \(3.\) So, \(333 \div 3 = 111.\) Now, \(111\) is also can be divided by \(3.\) So, \(111 \div 3 = 37.\) Here, \(37\) is a prime number so, \(37 \div 37 = 1.\)
Below is the division method to find the prime factors of \(999.\)

So, the prime factors of \(999\) are \(3 \times 3 \times 3 \times 37\) which can be written in exponential form as \({3^3} \times 37.\)

Q.5. Find the prime factors of \(544\) using the factor tree method.
Ans: We know that the first prime number which divides \(544\) is \(2.\) So, \(544 \div 2 = 272.\) Also, \(272\) is divided by \(2.\) So, \(272 \div 2 = 136.\) Now, \(136\) is also can be divided by \(2.\) So, \(136 \div 2 = 68.\) Now, \(68\) can also divisible by \(2.\) So, \(68 \div 2 = 34.\) Also, \(34 \div 2 = 17\) and \(17\) is a prime number. Hence \(17 \div 17 = 1\)
Below is the factor tree method to find the prime factors of \(544.\)

So, the prime factors of \(544\) are \(2 \times 2 \times 2 \times 2 \times 2 \times 17\) and its exponential form is \({2^5} \times 17.\)

Summary

From the above article, we can conclude that we have gained knowledge about the prime numbers, factors and the methods to find the prime factors. Also, we have learned the prime factorization and methods of prime factorization and some solved examples using both types of prime factorization.

Frequently Asked Questions (FAQ) – Prime Factorisation

Q.1. What is the prime factorization of \(40?\)
Ans: We use the easiest method, that is division method to find the prime factors of \(40.\)

Q.2. How do you find prime factorization?
Ans : Prime factorization is the process of writing a number as the product of prime numbers. To find the prime factors of a number one should divide the number by prime numbers until the remainder is equal to \(1.\)
We use two methods to find the prime factorization. They are,
a. Division method
b. Factor tree method

So, the prime factors of \(40\) are \(2 \times 2 \times 2 \times 5\) and its exponential form is \({2^3} \times 5.\)

So, the prime factors of \(12\) are \(2 \times 2 \times 3\) and its exponential form is \({2^2} \times 3.\)

Q.3. What is the prime factorization of \(24.\)
Ans: Prime factorization of \({\rm{24}}\) using the division method is shown below.

So, the prime factors of \(12\) are \(2 \times 2 \times 2 \times 3\) and its exponential form is \({2^3} \times 3.\)

Q.4. What is the prime factorization of \(12?\)
Ans : Prime factorization of \(12\) using the division method is shown below.

Q.5. What is the prime factorization of \(729.\)
Ans : Prime factorization of \(729\) using the division method is shown below.

So, the prime factors of \(729\) are \(3 \times 3 \times 3 \times 3 \times 3 \times 3\) and its exponential form is \({3^6}.\)

Q.6. What is the prime factorization of \(256?\)
Ans : Prime factorization of \(256\) using the division method is shown below.

So, the prime factors of \(256\) are \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\) and its exponential form is \({2^8}.\)

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