Question

What is the least number by which $ 2352 $ is to be multiplied to make it a perfect square?
(A) $ 6 $
(B) $ 4 $
(C) $ 3 $
(D) $ 8 $

Answer 1

Hint: To find the required least number, we will use the prime factorization method. We will write the given number $ 2352 $ as the multiple of primes. After that it will be written in the form of a group of two primes if possible.

Complete step-by-step answer:
To solve the given problem, we must know the prime factorization method. By using the method of prime factorization, we can express the given number as a product of prime numbers. Therefore, we will write the given number $ 2352 $ as the product of primes. Let us do the prime factorization of $ 2352 $ . Note that $ 2352 $ is an even number so we can start prime factorization with prime number $ 2 $ .

$ 2 $	$ 2352 $
$ 2 $	$ 1176 $
$ 2 $	$ 588 $
$ 2 $	$ 294 $
$ 3 $	$ 147 $
$ 7 $	$ 49 $
$ 7 $	$ 7 $
	$ 1 $

Therefore, we can write $ 2352 = 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7 $ . Here we are dealing with a perfect square so we have to write the obtained factorization in the form of a group of two primes if possible. So, we can write $ 2352 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times 3 \times \left( {7 \times 7} \right) \cdots \cdots \left( 1 \right) $ . To find a perfect square root, we have to take one number from each group of two but here we can see that a single $ 3 $ cannot be written as a group of two.
Let us multiply by $ 3 $ on both sides of the equation $ \left( 1 \right) $ . So, we can write
 $ 2352 \times 3 = \left[ {\left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times 3 \times \left( {7 \times 7} \right)} \right] \times 3 $
 $ \Rightarrow 2352 \times 3 = \left( {2 \times 2} \right) \times \left( {2 \times 2} \right) \times \left( {3 \times 3} \right) \times \left( {7 \times 7} \right) $
Here we can take one number from each group of two. So we will get the perfect square root. Hence, the required least number is $ 3 $ . Hence, option C is correct.
So, the correct answer is “Option C”.

Note: Remember that if the number is even then it is divisible by $ 2 $ . If the sum of all digits of a number is divisible by $ 3 $ then that number is also divisible by $ 3 $ . Double the last digit of the number and subtract the doubled number from the remaining number (remaining digits). If the result is divisible by $ 7 $ then that number is divisible by $ 7 $ . Note that here we will consider positive differences. In the given problem, $ 49 $ is divisible by $ 7 $ because double of last digit $ 9 $ is $ 18 $ and positive difference of $ 18 $ and remaining number (remaining digit) $ 4 $ is $ 14 $ and the number $ 14 $ is divisible by $ 7 $ .