How do you find square root of $361?$

Hint:
Whenever we need to find the square root of any given number, first we need to know or we need to find its prime factors. After finding a suitable prime factor if we multiply then we need to get the same number.

Complete Step by Step Solution:
Whenever we need to find the square root of any given number, first we need to know or we need to find its prime factors.
Prime numbers are nothing but the numbers which can be divided by $1$ and the number itself. The starting prime numbers are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31$ and so on.
Now, we check for prime factor:
The given number is $361$, so it is not an even number. The sum of the digits are not multiple of $3$.
Whenever we try to compare multiples of $5$, the last digit of the number has to be $0$ or $5$, so the number is not a multiple of $5$ as the number’s last digit is $1$.
Now, we try for the next prime number that is $7$. If we divide the number $361$ with $7$ it will give the remainder as $4$. So, it is not a prime factor.
Point to be remembered: A prime factor is said to be a number which gives zero as the remainder.
Now, if we take the next prime number that is $11$, this will give us the remainder as $9$. So we check for the next prime number that is $13$, which will give $10$ as remainder.
Next prime number $17$ gives $4$ as remainder. While the next prime number $19$gives zero as remainder so the required prime factor is $19$.
To do cross verification, we get as below
$19 \times 19 = 361$

Hence the square root of the number $361$ is $19$.

Note:
The square root can also be found by using a long division method as well. Both the prime factor and long division method gives the correct answer. Only thing is by using the prime factor method you need to know prime numbers to check whether it will divide the number properly or not.