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Prime Factorization – Meaning, Methods & Examples

prime factorization

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A prime number is a number that has only two factors – $1$ and the number itself. A number that is not a prime number is called a composite number. Any composite number can be expressed as a product of prime numbers. Prime factorization is a way of expressing a number as a product of its prime factors. 

For example, a composite number $6$ can be expressed as $2 \times 3$, where both $2$ and $3$ are prime numbers. Therefore, prime factorization of $6$ is $2 \times 3$.

What is Prime Factorization?

The method of breaking down a number into its prime numbers that help in forming the number when multiplied is called prime factorization. In other words, when all prime factors are multiplied together, you get the number whose prime factorization is obtained.

For example, prime factors of $48$ are $2$, $2$, $2$, $2$, and $3$, as $2 \times 2 \times 2 \times 2 \times 3 = 48$, or $\2^{4} \times 3 = 48$.

Similarly, prime factors of $90$ are $2$, $3$, $3$, and $5$ since, $2 \times 3 \times 3 \times 5 = 90$, or $2 \times 3^{2} \times 5 = 90$.

Difference Between Factors and Prime Factors

The factors of a number are the numbers that divide a given number completely without leaving any remainder. For example $3$ and $6$ and factors of $18$, since $3 \times 6 = 18$. Similarly, $2$ and $9$ are factors of $18$, as $2 \times 9 = 18$.

The prime factors of a number are the prime numbers that when multiplied give the original number. For example,  $2$, $3$, and $3$ are the prime factors of $18$, since, $2 \times 3 \times 3 = 18$.

Factors of $18$Prime Factors of $18$
Factors of $18$ are $1$, $2$, $3$, $6$, $9$, and $18$Prime factors of $18$ are $2$, $3$ and $3$, since $2 \times 3 \times 3 = 2 \times 3^{2} = 18$.

Note 

  • A factor of a number can be a prime number or a composite number, whereas, a prime factor of a number is always a prime number.
  • All prime factors of a number are factors of a given number, whereas, all factors of a given number are not prime factors.

Methods of Prime Factorization

There are various methods for the prime factorization of a number. The most common methods that are used for prime factorization are 

  • Prime factorization by factor tree method
  • Prime factorization by division method

Let’s understand how these methods are used to find the prime factorization of a number.

Prime Factorization by Factor Tree Method

In the factor tree method, you always start with the smallest factor of a number that is prime also and split the number. The resultant number is again in terms of a prime factor and a number. The process is repeated until the number cannot be split further.

Steps of Prime Factorization by Factor Tree Method

Let’s consider the number $840$ to understand the steps involved in the factor tree method of prime factorization. 

Step 1: Place the number, $840$, on top of the factor tree

Step 2: Then, write down a pair of factors as the branches of the tree starting with the lowest possible prime number. Here, they are $2$ and $420$

Step 3: Repeat the process for $420$. The factor pair is $2$ and $210$

Step 4: Repeat the process for $210$. The factor pair is $2$ and $105$

Step 5: Repeat the process for $105$. The factor pair is $3$ and $35$

Step 6: Repeat the process for $35$. The factor pair is $5$ and $7$

Step 7: Both factors in step 6 are prime numbers, so this completes the process

prime factorization

Let’s consider one more example of a number $48$.

prime factorization
Maths in Real Life

Prime Factorization by Division Method

The division method can also be used to find the prime factors of a large number by dividing the number by prime numbers. Let’s learn how to find the prime factors of a number by the division method using the number $72$.

Steps of Prime Factorization by Division Method

Step 1: Divide the number by the smallest prime number such that the smallest prime number should divide the number completely. Here we divide $72$ by $2$ to get $36$

Step 2: Again, divide the quotient of step 1 by the smallest prime number.  $36$ is again divided by $2$ and we get $15$

Step 3: Repeat step 2, until the quotient becomes $1$

Step 4: Finally, multiply all the prime factors that are the divisors. Prime factorization of $72 = 2 \times 2 \times 3 \times 3$

prime factorization

Let’s consider one more example of a number $252$.

prime factorization

Where Prime Factorization is Used?

Prime factorization is used to find the H.C.F. and L.C.M. of numbers. 

The Highest Common Factor (HCF) of two numbers is the highest possible number that divides both the numbers completely. The Highest Common Factor (HCF) is also called the Greatest Common Divisor (GCD).

The Least Common Multiple (LCM) of two or more numbers is the smallest number among all common multiples of the given numbers.

Conclusion

Prime factorization is a way of representing a number as a product of its prime factors. Prime factorization is used to find the H.C.F. and L.C.M. of numbers.

Practice Questions

Find the prime factorization of the following numbers

  • $36$
  • $48$
  • $56$
  • $68$
  • $78$
  • $86$
  • $122$
  • $156$
  • $272$

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FAQs

What is prime factorization in math?

Prime factorization of any number means to represent that number as a product of its prime factors. A prime number is a number that has exactly two factors, $1$ and the number itself. 

For example, the prime factorization of $48 = 2 \times 2 \times 2 \times 2 \times 3$. Here $2$ and $3$ are the prime factors of $48$.

How to do prime factorization?

Prime factorization of any number can be done by using any of these two methods:

Division method: In this method, the given number is divided by the smallest prime number which divides it completely or leaves the remainder $0$ on dividing. After this, the quotient is again divided by the smallest prime number. This step is repeated until the quotient becomes $1$ and cannot be divided further. Then, all the prime factors are collected and multiplied.

Factor tree method: In this method, the given number is placed on top of the factor tree. Then, the corresponding pairs of factors are written as the branches of the tree. After this step, the composite factors are again factorized and written down as the next branches. This procedure is repeated until we get the prime factors of all the composite factors.

Why is prime factorization important?

Prime factorization is used to find the Highest Common Factor (H.C.F.)  and the Lowest Common Multiple (L.C.M.) of numbers.

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