NCERT Class 8 Mathematics Ganita Prakash Part-I: Chapter 1 — A Square and A Cube
This chapter, "A SQUARE AND A CUBE," introduces the concepts of square numbers and their properties through an engaging locker puzzle. Queen Ratnamanjuri's will sets up a challenge for her son Khoisnam and 99 relatives involving 100 lockers. Each person, numbered 1 to 100, toggles lockers corresponding to their number. Khoisnam realizes that only lockers with a square number as their number will remain open at the end. This is because square numbers have an odd number of factors, meaning they are toggled an odd number of times. The chapter explains factors, partner factors, and how perfect squares have a unique factor that pairs with itself, resulting in an odd count of total factors. This mathematical insight helps students understand the nature of square numbers and their unique characteristic of having an odd number of divisors, crucial for CBSE Mathematics learning.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 8 |
| Subject | Mathematics |
| Book | Ganita Prakash Part-I |
| Chapter | Chapter 1 — A Square and A Cube |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 4 minutes |
| Word count | 655 |
Learning outcomes
- Understand the concept of square numbers.
- Identify numbers with an odd number of factors.
- Relate the number of factors to the toggling of lockers in a puzzle.
- Determine which lockers remain open based on their number being a perfect square.
- Grasp the relationship between perfect squares and an odd number of factors.
Vocabulary
| Word | Meaning |
|---|---|
| Ratnas | Precious stones |
| Toggles | Opens if closed, closes if open |
| Factors | Numbers that divide a given number without a remainder |
| Partner factor | A factor that, when multiplied by another factor, gives the original number |
| Square number | A number that is the product of an integer with itself (e.g., 4 = 2x2) |
| Odd number of factors | A count of factors that is not divisible by two |
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Practice questions
- Which lockers will remain open at the end of the puzzle? Answer: Lockers whose numbers are perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100).
- Explain why locker #9 will be open at the end. Answer: Locker #9 is open because 9 is a perfect square (3x3). Its factors are 1, 3, and 9 (an odd number of factors), meaning it's toggled an odd number of times.
- List the factors of the number 12. Answer: The factors of 12 are 1, 2, 3, 4, 6, and 12.
- Does the number 12 have an odd or even number of factors? How does this affect its locker? Answer: The number 12 has an even number of factors (6 factors). Therefore, locker #12 will be closed at the end.
- What is a square number? Answer: A square number is a number that can be expressed as the product of an integer with itself, like 4 (2x2) or 9 (3x3).
Frequently asked questions
What is the locker puzzle about?
The puzzle involves 100 lockers and 100 people. Each person toggles lockers based on their number, and the final state of the lockers reveals a code.
How does the number of factors relate to the locker's final state?
A locker remains open if its number has an odd number of factors, as it gets toggled an odd number of times. It remains closed if it has an even number of factors.
What kind of numbers have an odd number of factors?
Only perfect square numbers have an odd number of factors.
Why do perfect squares have an odd number of factors?
Perfect squares have one factor that, when multiplied by itself, equals the number (e.g., 3x3=9). This 'self-paired' factor results in an odd total count of factors.
Which lockers will be open at the end of the puzzle?
The lockers numbered with perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) will remain open.
What is the significance of the number 1 in this puzzle?
Locker #1 is open because 1 is a perfect square (1x1) and has only one factor (1), which is an odd number.
Related resources
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Topics covered
NCERT Class 8 Mathematics — Ganita Prakash Part-I — Chapter 1 — A Square and A Cube. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.