Important Questions Class 8 Maths Ganita Prakash – I Chapter 1 A Square and A Cube

Short Questions

1. What is a perfect square?

Answer: A number obtained by multiplying a number by itself.
Solution:
Example: $4 = 2 \times 2$, $9 = 3 \times 3$

2. Which numbers remain open in the locker problem?

Answer: Square numbers
Solution:
Only numbers with odd number of factors remain open → perfect squares like 1, 4, 9, 16…

3. Why do square numbers have an odd number of factors?

Answer: Because one factor is repeated
Solution:
Example: 36 → (6 × 6), so one factor has no pair

4. What is the square of 15?

Answer: 225
Solution:
$15 \times 15 = 225$

5. Which digits can appear at the unit place of a square number?

Answer: 0, 1, 4, 5, 6, 9
Solution:
Squares never end in 2, 3, 7, or 8

6. What is the sum of first 4 odd numbers?

Answer: 16
Solution:
$1 + 3 + 5 + 7 = 16 = 4^2$

7. What is the square root of 64?

Answer: 8
Solution:
$8 \times 8 = 64$

8. Is 156 a perfect square?

Answer: No
Solution:
Prime factorisation: $2 \times 2 \times 3 \times 13$ → cannot be paired equally

9. What is a perfect cube?

Answer: A number obtained by multiplying a number three times
Solution:
Example: $8 = 2 \times 2 \times 2$

10. What is the cube of 5?

Answer: 125
Solution:
$5 \times 5 \times 5 = 125$

Long Questions

1. Explain why only square-numbered lockers remain open.

Answer:
Lockers are toggled based on their factors.

Solution:

• Each locker is toggled once for every factor
• If toggled odd times → open
• If even → closed
• Only square numbers have odd number of factors
  Therefore, lockers numbered 1, 4, 9, 16, 25… remain open

2. Show that 25 is a perfect square using odd numbers.

Answer:
25 is a perfect square

Solution:
Subtract consecutive odd numbers:
25 – 1 = 24
24 – 3 = 21
21 – 5 = 16
16 – 7 = 9
9 – 9 = 0

Since it reaches 0 → perfect square
Also, $25 = 5^2$

3. Find whether 324 is a perfect square using prime factorisation.

Answer: Yes, it is a perfect square

Solution:
$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3$
Group:
$(2 \times 3 \times 3) \times (2 \times 3 \times 3)$

So,
$\sqrt{324} = 18$

4. Estimate the square root of 250.

Answer: Approximately 16

Solution:
$15^2 = 225$
$16^2 = 256$

Since 250 is closer to 256:
$\sqrt{250} \approx 16$

5. Show that 3375 is a perfect cube.

Answer: Yes, it is a perfect cube

Solution:
$3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5$

Group into triplets:
$(3 \times 5)^3$

So,
$\sqrt[3]{3375} = 15$