Finding Square Root Through Prime Factorization and Repeated Subtraction

Square Root is the one of the many arithmetic operations in mathematics. Square root can be calculated using various methods in mathematics such as long division, prime factorization, repeated subtraction, etc. In this article, we will discuss methods of calculation of square root using prime factorization and repeated subtraction method.

What are Square Roots?

Square root of a number is a value that, when multiplied by itself, gives the original number. It is represented using the symbol ‘√’. The number under the square root symbol is called the radicand.

The square root of a number can be found by looking for a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by itself equals 9.

Some Common Square Roots

Some common square root values are:

Number	Square Root
1	1
2	1.414
3	1.732
4	2
5	2.236
6	2.449
7	2.646
8	2.828
9	3
10	3.162

How to Find the Square Root of Any Number?

There are several methods for finding the square root of a number, including

• Repeated Subtraction Method
• Prime Factorization Method
• Estimation Method
• Long Division Method

In this article, we will discuss two common methods i.e., Repeated Subtraction Method and Prime Factorization Method in detail with solved examples.

Square Root by Prime Factorization

Prime factorization is the process of finding the prime factors of a number, which are the prime numbers that multiply together to make the original number.

To find the square root of a number using prime factorization, you can follow these steps:

Step 1: Prime Factorization: Decompose the given number into its prime factors.

Step 2: Pairing Factors: Pair up identical prime factors in twos.

Step 3: Square Roots of Each Pair: Take one factor from each pair and multiply them together.

Step 4: Multiply Results: Multiply the results from step 3 to find the square root of the original number.

Let’s consider an example for better understanding.

Example: Find square root of 3600 using prime factorization.

Solution:

Step 1: Prime Factorization of 3600:

3600 = 2⁴ × 3² × 5²

Step 2: Pairing Factors:

3600 = (2 × 2) × (2 × 2) × (3 × 3) × (5 × 5)

Step 3: Square Roots of Each Pair:

Take one factor from each pair: 2 × 2 × 3 × 5 = 60

Step 4: Multiply Results:

√3600 = 60

So, the square root of 3600 is 60.

Square Root by Repeated Subtraction

The repeated subtraction method is an easier way that uses a special property of odd numbers. When you add up consecutive odd numbers, starting from 1, you always get a perfect square.

Using this property of consecutive odd numbers, we can check whether the given number is perfect square or not. To check for any number, we can use the following steps:

Step 1: Start with the given number.

Step 2: Subtract consecutive odd numbers.

Step 3: Count the number of times you subtract.

Step 4: The count is the square root.

Let’s illustrate this with an example:

Example: Find the square root of 81 by repeated subtraction.

Solution:

Start with the given number: 81.

Subtract consecutive odd numbers starting from 1 until reaching zero:

1. 81 − 1 = 80
2. 80 − 3 = 77
3. 77 − 5 = 72
4. 72 − 7 = 65
5. 65 − 9 = 56
6. 56 − 11 = 45
7. 45 − 13 = 32
8. 32 − 15 = 17
9. 17 − 17 = 0

Count the number of times you subtracted an odd number: 9.

The square root of 81 is the number of times you subtracted, which is 9.

So, the square root of 81 is 9.

Which is the Quickest Method?

Quickest method for prime factorization depends on the number you’re trying to factorize and personal preference. However, in many cases, using the repeated subtraction method can be quicker for smaller numbers, especially when you’re dealing with numbers that have small prime factors.

Conclusion

Next time you want to find a square root, think about what you need. If it’s a perfect square and you want a quick answer, try repeated subtraction. But if you also want to know about the prime factors and understand the number better, go for prime factorization. Both methods show how interesting numbers can be and how they work in different ways.

Read More,

• Square Root of an Integer
• Squares and Square Roots
• Root 2 Value

Solved Example on Prime Factorization and Repeated Subtraction Method

Example: Find the square root of 7056 using prime factorization.

Solution:

7056 = 2⁴ × 3² × 7²

⇒ 7056 = (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7)

Take one factor from each pair: 2 × 2 × 3 × 7 = 84

Thus, √7056 = 84

So, the square root of 7056 is 84.

Example: Find the square root of 144 using prime factorization.

Solution:

144 = 2⁴ × 3²

⇒ 144 = (2 × 2) × (2 × 2) × (3 × 3)

Take one factor from each pair: 2 × 3 = 6

Thus, √144 = 12

So, the square root of 144 is 12.

Example: Find the square root of 4900 using prime factorization.

Solution:

4900 = 2² × 5² × 7²

⇒ 4900 = (2 × 2) × (5 × 5) × (7 × 7)

Take one factor from each pair: 2 × 5 × 7 = 70

Thus, √4900 = 70

So, the square root of 4900 is 70.

Example: Find the square root of 121 by repeated subtraction.

Solution:

Start with the given number: 121.

Subtract consecutive odd numbers starting from 1 until reaching zero:

• 121 − 1 = 120
• 120 − 3 = 117
• 117 − 5 = 112
• 112 − 7 = 105
• 105 − 9 = 96
• 96 − 11 = 85
• 85 − 13 = 72
• 72 − 15 = 57
• 57 − 17 = 40
• 40 − 19 = 21
• 21 − 21 = 0

Count the number of times you subtracted an odd number: 11.

The square root of 121 is the number of times you subtracted, which is 11.

So, the square root of 121 is 11.

Practice Problems on Prime Factorization and Repeated Subtraction Method

Problem 1: Find the square root of 81?

Problem 2: Calculate the square root of 144?

Problem 3: Determine the square root of 225?

Problem 4: What is the square root of 400?

FAQs on Prime Factorization and Repeated Subtraction Method

What is Prime Factorization?

Prime factorization is the process of breaking down a composite number into its prime factors. A prime factor is a prime number that divides the original number evenly.

What is the Repeated Subtraction Method?

The repeated subtraction method is a technique used to find the prime factors of a number. It involves continuously subtracting the smallest prime number that divides the given number until the result is 1. The prime factors are the numbers that are subtracted.

What are Prime Numbers?

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11, and so on.

Can All Numbers be Prime Factorized?

Yes, every composite number (a number greater than 1 that is not prime) can be expressed as a unique product of prime factors. This is known as the fundamental theorem of arithmetic.