Question

Find the least number by which $ 384 $ must be divided to make it a perfect square.

Answer 1

Hint: First of all find the prime factorization of the given number and then make the group of the two similar numbers and the resultant number which is not paired and left is the smallest number by which the given number $ 384 $ should be divided to get the perfect square.

Complete step by step solution:
First find factors of the given number $ 384 $
Factorization by using prime factorization –
In prime factorization, the product of the prime numbers gives the original number. Start finding the factors of the given number with least prime number. $ $
Now, write the given number in the form of product of prime factors.
 $ 384 = 2 \times 192 $
Find the factors until all the numbers are prime in the expression.
 $
  384 = 2 \times 2 \times 96 \\
  384 = 2 \times 2 \times 2 \times 48 \\
  384 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \\
  $
Take a pair of the prime numbers on the right hand side of the equation.
 $ 384 = \underline {2 \times 2} \times \underline {2 \times 2} \times \underline {2 \times 2} \times 2 \times 3 $
By observing the above equation, we can make out that the numbers $ 2 $ and $ 3 $ are left and not paired with the other same number.
Therefore, to make the given number the perfect square, it must be divided by $ 2 $
Now,
 $ \sqrt {\dfrac{{384}}{{2 \times 3}}} = \sqrt {\dfrac{{\underline {2 \times 2} \times \underline {2 \times 2} \times \underline {2 \times 2} \times 2 \times 3}}{{2 \times 3}}} $
[As the same numbers from numerator and denominator cancels each other]
 $ \sqrt {\dfrac{{384}}{{2 \times 3}}} = \sqrt {\underline {2 \times 2} \times \underline {2 \times 2} \times \underline {2 \times 2} } $
Therefore, the required solution is – The smallest number by which $ 384 $ be divided is the number $ 6 $ , so that the result term is a perfect square.
So, the correct answer is “6”.

Note: Be careful while finding the prime factors of the given term. For that be good in multiples and division. The perfect square can be defined as the product of the two equal integers. For example: $ 9 $ , it can be expressed as the product of two equal integers. $ 9 = 3 \times 3 $ whereas the square-root is denoted by $ \sqrt {{n^2}} = \sqrt {n \times n} $ For Example: $ \sqrt {{2^2}} = \sqrt 4 = 2 $