Question

The square root of 53361 is:
(a) 231
(b) 211
(c) 261
(d) 249

Answer 1

Hint: To determine the square root of a number, express the number in terms of the product of its prime factors. This is the method of prime factorization. Once you express the number as the product of its prime factors, start making the pairs of prime factors and counting them once for each pair to find the square root.

Complete step-by-step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.

Now, coming to the question, we will use the method of prime factorization. So, first we will express 53361 as the product of prime numbers.
The sum of digits of the number 53361 is 5+3+3+6+1=18, which is divisible by 9, hence the number is divisible by 9 and 9 is the product of two 3s. so, 53361 can be written as $53361=3\times 3\times 5929$ . Now the number 59929 is odd so not divisible by 2, also the sum of the digits is not divisible by 3. So, the next prime in the list is 7 and 5929 is divisible by 7. Therefore, 53361 can be expressed as $53361=3\times 3\times 7\times 847$ and we know that 847 when divided by seven leaven remainder zero and quotient as 121. Also, 121 is the square of 11. So, 53361 can be written as :
$53361=3\times 3\times 7\times 7\times 11\times 11$ .
Now if we take square root of 53361, we get
 $\sqrt{53361}=\sqrt{3\times 3\times 7\times 7\times 11\times 11}$
$\Rightarrow \sqrt{53361}=3\times 7\times 11$
$\Rightarrow \sqrt{53361}=231$
Therefore, the answer to the above question is option (a).

Note: While calculating square roots and cube roots, prime factorization is the easiest method. But it takes time. Hence, other methods should also be learnt, so that they can be used while solving problems in cases where time plays an important role. Also, for finding roots of the decimal as well as those numbers which are not perfect squares, the division method is the only method that can be applied.