Question

Find the perfect square number between
i) $30$ and $40$
ii) $50$ and $60$

Answer 1

Hint: To find perfect square between two numbers.
Perfect square numbers are those numbers whose prime factors are written in the power of even terms that is \[b = {a^n} \times {c^m} \times .......\], where a & c are prime numbers and n & m are even numbers. So using this pattern we will try to find perfect square number between \[\left( i \right)30{\text{ }}and{\text{ }}40{\text{ }}\left( {ii} \right)50{\text{ }}and{\text{ }}60\]. So firstly we will factorise the number and then write it in index form.

Step by step solution:
For \[\left( i \right)between{\text{ }}30\,and{\text{ }}40\]
Now we will try to write all the numbers present between in terms of their prime factors.
\[\begin{array}{l}
31 = 31 \times 1{\text{ }}32 = {2^5}{\text{ }}33 = 3 \times 11\\
34 = 2 \times 17{\text{ }}35 = 5 \times 7{\text{ }}36 = {3^2} \times {2^2}\\
37 = 37 \times 1{\text{ }}38 = 2 \times 19{\text{ }}39 = 39 \times 1
\end{array}\]
From the above factorization it is clear that only \[36\] has prime factors which can be written in even powers. So, our perfect square number between \[30{\text{ }}and{\text{ }}40{\text{ }}is{\text{ }}36\].
\[\left( {ii} \right)for{\text{ }}numbers{\text{ }}between{\text{ }}50{\text{ }}and{\text{ }}60\]
We will try to factorise all the numbers present between \[50{\text{ }}and{\text{ }}60\]
\[\begin{array}{l}
51 = 3 \times 17{\text{ }}52 = {2^2} \times 13{\text{ }}53 = 1 \times 53\\
54 = 2 \times {3^3}{\text{ }}55 = 5 \times 11{\text{ }}56 = {2^3} \times 7\\
57 = 19 \times 3{\text{ }}58 = 29 \times 2{\text{ }}59 = 59 \times 1
\end{array}\]
As it can be noticed that none of the numbers between \[50{\text{ }}and{\text{ }}60\] is a perfect square. So, no perfect square is present between \[50{\text{ }}and{\text{ }}60\].

Note:
A prime number is never a perfect square number. We must write all prime factors carefully. One more method is that if we remember squares of some standard number then it will save our time and particularly in this type of question we need not to find prime factors of all numbers. So it is advisable to learn squares of numbers as much as possible.