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Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer? [Hint: Find out how many times each locker is toggled.]
Khoisnam realized that each locker would be toggled for every factor it has. Most numbers have even factors because they come in pairs. However, perfect squares have one repeated factor, making their count odd. Since an odd number of toggles leaves a locker open, only perfect square locker numbers (Read more
Khoisnam realized that each locker would be toggled for every factor it has. Most numbers have even factors because they come in pairs. However, perfect squares have one repeated factor, making their count odd. Since an odd number of toggles leaves a locker open, only perfect square locker numbers (like 1, 4, 9…) remain open. This insight helped Khoisnam determine which lockers would stay open without watching the whole process.
For more NCERT Solutions for Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube Extra Questions & Answer:
https://www.tiwariacademy.com/ncert-solutions/class-8/maths/ganita-prakash-chapter-1/
See lessCan you use this insight to find more numbers with an odd number of factors?
We can use the insight about factor pairs to find all numbers with an odd number of factors. Only perfect squares like 1 (1×1), 4 (2×2), 9 (3×3), etc., have a repeated factor, giving them an odd count. These square numbers are the only ones toggled an odd number of times, so only they remain open. TRead more
We can use the insight about factor pairs to find all numbers with an odd number of factors. Only perfect squares like 1 (1×1), 4 (2×2), 9 (3×3), etc., have a repeated factor, giving them an odd count. These square numbers are the only ones toggled an odd number of times, so only they remain open. This understanding helps in identifying perfect squares just by counting their factors.
For more NCERT Solutions for Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube Extra Questions & Answer:
https://www.tiwariacademy.com/ncert-solutions/class-8/maths/ganita-prakash-chapter-1/
See lessDoes every number have an even number of factors?
Most numbers have an even number of factors, as they come in factor pairs. For example, 6 has 1×6 and 2×3. However, perfect square numbers like 36 have one repeated factor (6×6), making the total number of factors odd. So, only perfect squares have an odd number of factors. This property is key to sRead more
Most numbers have an even number of factors, as they come in factor pairs. For example, 6 has 1×6 and 2×3. However, perfect square numbers like 36 have one repeated factor (6×6), making the total number of factors odd. So, only perfect squares have an odd number of factors. This property is key to solving the locker puzzle and understanding which lockers stay open.
For more NCERT Solutions for Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube Extra Questions & Answer:
https://www.tiwariacademy.com/ncert-solutions/class-8/maths/ganita-prakash-chapter-1/
See lessWhich are these five lockers?
The five lockers that are toggled exactly twice are locker numbers 2, 3, 5, 7 and 11. These are prime numbers and have exactly two distinct factors—1 and themselves. This means they are toggled only by person 1 and one other person (themselves), resulting in exactly two toggles. Khoisnam uses this kRead more
The five lockers that are toggled exactly twice are locker numbers 2, 3, 5, 7 and 11. These are prime numbers and have exactly two distinct factors—1 and themselves. This means they are toggled only by person 1 and one other person (themselves), resulting in exactly two toggles. Khoisnam uses this knowledge to extract the code clue from these specific lockers that were touched only twice during the puzzle.
For more NCERT Solutions for Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube Extra Questions & Answer:
https://www.tiwariacademy.com/ncert-solutions/class-8/maths/ganita-prakash-chapter-1/
See lessWrite the locker numbers that remain open.
Only lockers with numbers that are perfect squares remain open because they are toggled an odd number of times. The perfect square numbers from 1 to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Each of these has a repeated middle factor, making their total factor count odd. Hence, these 10 lockRead more
Only lockers with numbers that are perfect squares remain open because they are toggled an odd number of times. The perfect square numbers from 1 to 100 are: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Each of these has a repeated middle factor, making their total factor count odd. Hence, these 10 lockers are the only ones that stay open in the end.
For more NCERT Solutions for Class 8 Mathematics Ganita Prakash Chapter 1 A Square and A Cube Extra Questions & Answer:
https://www.tiwariacademy.com/ncert-solutions/class-8/maths/ganita-prakash-chapter-1/
See less