Question

Find the smallest number by which 2592 be multiplied so that the product is a
perfect square.
A) 2
B) 4
C) 5
D) 8

Answer 1

Hint: Here, we will find the smallest number by which the given number can be multiplied so that the product is a perfect square.

Complete step by step answer:
  To find the perfect number first we will find the factors of this given number 2592 using prime factorization method.
So, the prime factors of 2592 is;
\[\text{factorisation of }2592=2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3\]

On grouping the prime factors in pairs we see that one 2 is left out…..
\[\text{ }2592={{2}^{2}}\times {{2}^{2}}\times {{2}^{2}}\times 2\times {{3}^{2}}\times {{3}^{2}}\]

Hence, we should multiply with 2, so that prime factor 2 gets grouped in pairs.
And since, according to the question we were told to find the smallest number by which 2592 should be multiplied so that it can be made a perfect square then the answer should be 2.

We did the pairing since we need to find the perfect square of the number 2592. Or we could have simply tried to find the square root of 2592 , (\[\sqrt{2592}\]), since finding the square root also means to find its perfect square if no decimal number is obtained.
Thus,
\[\sqrt{2592}=\sqrt{2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3}=36\sqrt{2}\]
 We see that root 2 is left out which means we need one more root 2 to make it a perfect square.

Thus the answer of this question is option A.

Note: In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.
For example, 9 is a square number, since it can be written as 3 × 3. Thus either we do \[\sqrt{9}\] which will lead us to 3 or we take out 9’s prime factor and try to find the perfect root through that method we get the same answer , that is, 3