How to you find the square root of composite numbers?
Answers
Step-by-step explanation:
To check whether a given number N is prime or not, first find the square root of that number N and then approximate that to immediately lower integer (say n) and write down all the prime numbers less than that integer (n). Then check the divisibility of the given number N by all the prime numbers we have written in previous step, if it is not divisible by any of the prime numbers then given number N is prime.
Let us write algorithm for the same
Step 1: Find square root of N, call it as K (Just find approximate values)
Step 2: Write down all the prime numbers less than K.
Step 3: Check divisibility of N with these prime numbers, which we have got in Step 2.
Step 4: If N is not divisible by any of the prime numbers then N is prime.
Example:
Let us check whether 211 is prime or not?
Solution:
Step 1: We find square root of 211 i.e. K=√211 = 14.52
Step 2: We write all primes less than 14.52 i.e. 2, 3, 5, 7, 11 and 13.
Step 3:Since 211 is not divisible by any of these prime numbers, hence 211 is a prime number.
To solve this question, we will use the concept of prime factorization. The fundamental theorem of arithmetic states that every composite number can be expressed or we can say that it is factorized as a product of prime numbers, and this factorization is unique except for the order in which the prime factors occur.
Complete step-by-step answer:
The numbers that are divisible by only two numbers that are 1 and by themselves are called prime numbers.
Given that, 1936.
We have to find its square root by a prime factorization method.
First, we have to make the prime factors of 1936.
⇒1936=2×968⇒1936=2×2×484⇒1936=2×2×2×242⇒1936=2×2×2×2×121
⇒1936=2×2×2×2×11×11
………… (i)
These are the prime factors of 1936.
Now, we will find out its square root.
Taking square root on both sides of equation (i), we will get
⇒1936−−−−√=2×2×2×2×11×11−−−−−−−−−−−−−−−−−−√⇒1936−−−−√=(2×2)×(2×2)×(11×11)−−−−−−−−−−−−−−−−−−−−−−−√⇒1936−−−−√=(2×2)2×(11)2−−−−−−−−−−−−−√⇒1936−−−−√=2×2×11⇒1936−−−−√=44
Here, we can see that the square root of 1936 by the prime factorization method is 44.
Note: Whenever we ask this type of question, first, we have to remember what are prime numbers? And also, we should know the prime factorization method. Then we have to find out the prime factors of the given number and after that we will do the square root of that number and its prime factors. By solving this, we will get the required answer.