Question

Find the least number by which \[2888\] should be multiplied to obtain a perfect square. Also, find the square
root of the number so obtained.
A. \[2,{\text{ }}76\]
B. \[10,{\text{ }}76\]
C. \[4,{\text{ }}76\]
D. \[1,{\text{ }}76\]

Answer 1

Hint: Firstly, check the given number whether it is a perfect square or not, if not then multiply it with suitable number to
get the perfect square.


Complete step by step solution:
1. Given number is $2888$. We first make prime factors of \[2888\]

2. Prime factors of $2888$ are $2 \times 2 \times 2 \times 19 \times 19$. Clearly we see that there is only one number i.e. $2$ which is not in pair.
Therefore, we can say that $2888$ is not a perfect square number.
3. Now, to make \[2888\] a perfect square number, we can either multiply or divide it with some number but as in the
statement we have asked to multiply \[2888\] to make it a perfect square. So, we chose a number such that pair in
prime factors of \[2888\] get completed.
4. So, it is very clear that the number is \[2\] by which multiplying $2888$, prime factors get paired
i.e. $\underbrace {2 \times 2}_{} \times \underbrace {2 \times 2}_{} \times \underbrace {19 \times 19}_{}$
5. Hence, from above we can say that \[2\]is the least number by which $2888$ gets multiplied to make it a perfect square.
6. Now, to find the square root of the resulting number so obtained.
As we can see in step \[4\] after multiplying $2888$ by $2\left( {5776} \right)$ pairs of prime factors get completed.
Hence, number obtained on multiplying will be a perfect square number and its square root will be given as
$
  \sqrt {2 \times 2 \times 2 \times 2 \times 19 \times 19} \\
   = 2 \times 2 \times 19 \\
   = 76 \\
 $
7. Hence, from above we see that \[2\]is the smallest number by which \[2888\] needs to multiply to get a perfect square the number and square root of the number so obtained is \[76\].
So, the correct option is A.


Note: If there is a perfect square number, then we will have to take each prime factor even time i.e. $2$ should be taken $4$ times,$19$ should be taken $2$ times