Question

The square root of 390625 is:
(a) 645
(b) 225
(c) 625
(d) 635

Answer 1

Hint: Remember the point that the number ending with the digit 5 is always divisible by 5. Now 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.
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 390625 as the product of prime numbers.
The unit digit of the number 390625 is 5, so it is definitely divisible by 5. So, it can be written as $390625=78125\times 5$ . The unit digit of 78125 is also 5, which means it is also divisible by 5. Therefore, $390625=15625\times 5\times 5$ . Again, 15625 is divisible by 5, so $390625=3125\times 5\times 5\times 5$ .
Repeating the same steps, i.e., checking the divisibility with 5, we get $390625=5\times 5\times 5\times 5\times 5\times 5\times 5\times 5$
Now if we take square root of 390625, we get
 $\sqrt{390625}=\sqrt{5\times 5\times 5\times 5\times 5\times 5\times 5\times 5}$
$\Rightarrow \sqrt{390625}=5\times 5\times 5\times 5$
$\Rightarrow \sqrt{390625}=625$
Therefore, the answer to the above question is option (c).

Note: For finding the square of any number ending with 25, you can use a short trick. Let the last 3 digits of your answer be 625. Then find the square of the number with the highest place in the number of which we are finding the square and divide the same digit by 2 and add the two results, i.e., the square and the half of the first digit and multiply the result by 10. The number you get is the first digits of the square. For example: let us find the square of 625. So, the last digits of our answer are 625. Also, the half of 6 is 3 and square of 6 is 36 and the sum of these two are 39 which when multiplied by 10 gives 390 as the result which is the first digits of our answer. So, the answer is 390625.