Q: What is the value of $\sqrt{625}$?

By prime factorisation of 625: $$625 = 5 \times 5 \times 5 \times 5$$ $$\sqrt{625} = 5 \times 5 = 25$$

Question 1

What is the unit digit of the square of the number 67?

Accepted Answer

To find the unit digit of $67^2$, we look at the unit digit of $67$, which is $7$. 
Since $7^2 = 49$, the unit digit is 9.

Question 2

How many non-square numbers lie between $12^2$ and $13^2$?

Accepted Answer

The number of non-square numbers between $n^2$ and $(n+1)^2$ is given by the formula $2n$. 
Here $n = 12$, so: $$	ext{Number of non-square numbers} = 2 	imes 12 = 24$$

Question 3

What is the sum of the first 8 odd natural numbers?

Accepted Answer

The sum of the first $n$ odd natural numbers is given by $n^2$. 
Here $n = 8$: $$	ext{Sum} = 8^2 = 64$$

Question 4

Which of the following is a Pythagorean triplet whose smallest member is 6?

Accepted Answer

A Pythagorean triplet is of the form $(2m, m^2 - 1, m^2 + 1)$. 
Given smallest member $2m = 6 \implies m = 3$. Second member: $m^2 - 1 = 3^2 - 1 = 8$ Third member: $m^2 + 1 = 3^2 + 1 = 10$ 
Thus, the triplet is (6, 8, 10).

Question 5

What is the value of $\sqrt{625}$?

Accepted Answer

By prime factorisation of 625: $$625 = 5 	imes 5 	imes 5 	imes 5$$ $$\sqrt{625} = 5 	imes 5 = 25$$

Square and Square Roots