Question

What least number must be multiplied to 12288 so that the product becomes a perfect square?
A.2
B.3
C.4
D.5

Answer 1

Hint: To solve this type of question first of all we have to break the given number in prime factorisation form, then find which prime factor is not in pair or we can say having odd number of prime factors. That will be the answer to the required question.

Complete step-by-step answer:
The given number is 12288.
We have to find which number should be multiplied with this number such that it can be converted into a perfect square number.
To get that number which makes 12288 is a perfect square can be found by finding the prime factors of the given number.
\[\Rightarrow 12288 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\]
In the above prime factorisation of the number 12288 we see that 3 is the only prime factor which comes single time or we can say that odd number of times.
So if we multiply the given number by 3 the number will have all paired prime factors or we can say even number of prime factors.
So it is clear we need one more 3 to multiply to make perfect square
∴\[12288 \times 3 = \sqrt {36864} = 192\]
Hence the required number is 3.
So, the correct answer is “3”.

Note: The number which is already in perfect square form can also be converted in perfect square by multiplying the given number by 1, while you have to remember that 1 is not a prime number. The least prime number 2.