Question

Perfect square number between 20 and 30 is/are ?

Answer 1

Hint: In order to determine the perfect square numbers between $$20$$ and $$30$$. First, we have to list down all the numbers between $$20$$ and $$30$$ then we need to check the perfect square numbers among them. Now, we have to eliminate those numbers which are prime numbers. If it is expressible as a product of two same numbers it is a perfect square.

Complete step by step solution:
In this problem, we need to analyse the perfect square number between $$20$$ and $$30$$.
First, we have to list down all the number between $$20$$ and $$30$$:
$$21,22,23,24,25,26,27,28$$ and $$29$$ --------(1)
We know that the perfect square numbers of units digit always end with $$0,1,4,5,6,9$$.
From equation (1),
We have to check whether the number of unit digits ends with $$0,1,4,5,6,9$$ or not.
Therefore, the numbers between $$20$$ and $$30$$ end with unit digits are:
$$21,24,25,26$$ and $$29$$.
Now, we will check these numbers one by one for a perfect square.
$$21=3\times 7$$ , $$24=2\times 12$$ , $$25=5\times 5$$ , $$26=2\times 13$$ and $$29$$ is a prime number, not a perfect square.
Thus, we can see that the number $$25$$ is expressible as a product of prime numbers in pairs i.e. $$5\times 5$$.
Hence,$$25$$ is the perfect square number between $$20$$ and $$30$$.

Note:
Perfect square number is a number which is a square of some number.
Since, when some number is multiplied by itself we obtain a number that is a perfect square .
For example:
$$\begin{gathered} 5^2=5\times 5=25 \\ {7}^2=7\times 7=49 \end{gathered}$$ where $$5$$ and $$49$$ are perfect square numbers.
Perfect square numbers always end in (i.e. unit digit) $$0,1,4,5,6$$ and $$9$$.
Prime numbers are those numbers which are either divisible by $$1$$ and itself. Prime numbers can never be perfect squares. For example $$29$$ is a prime number which is not a perfect square.