Question

What is the least number by which 300 is to be multiplied so that the number obtained is a perfect square? Also, find the square root of the resulting number?

Answer 1

Hint: To solve this problem you should know the factorization method. In the factorization method we write any number in the form of multiplication of its prime factors. Then we may group these prime factors as pairs to find which prime factors do not have pairs. That would be the required number to be multiplied in order to make it a perfect square.

Complete step-by-step solution:
Writing 300 in the form of multiplication of its prime numbers, we get
$300 = 2 \times 2 \times 5 \times 5 \times 3$
We can see that 3 remains without a pair. Hence we need to multiply 300 by 3 in order to make it a perfect square.
Multiplying 300 by 3, we get
$300 \times 3 = 2 \times 2 \times 3 \times 3 \times 5 \times 5$
$900 = {2^2} \times {3^2} \times {5^2}$
Now, taking square root of both sides, we get
$\sqrt {900} = \sqrt {{2^2} \times {3^2} \times {5^2}} $
$\sqrt {900} = 30$
Therefore, the square root of the required number is 30.

Note: In these types of questions in which you need to find the square root of any number, firstly you should write that number in the form of multiplication of its prime factors and then make pairs of the same prime factors. After that take one number from each pair and multiply them all. The resulting number you will get, would be the required square root.