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Does 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 lessWhat patterns do you notice? Share your observations and make conjectures.
We observe that square numbers always end with digits 0, 1, 4, 5, 6 or 9—never 2, 3, 7 or 8. Square numbers are formed by summing consecutive odd numbers starting from 1. Another pattern is that only square numbers have an odd number of factors. These patterns help in identifying whether a number isRead more
We observe that square numbers always end with digits 0, 1, 4, 5, 6 or 9—never 2, 3, 7 or 8. Square numbers are formed by summing consecutive odd numbers starting from 1. Another pattern is that only square numbers have an odd number of factors. These patterns help in identifying whether a number is a perfect square and predicting how many toggles lockers undergo in the puzzle logic.
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 lessIf a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?
Even though square numbers always end in 0, 1, 4, 5, 6 or 9, not all numbers ending in these digits are squares. For instance, 36 and 16 are squares ending in 6, but 26 also ends in 6 and is not a square. So, we cannot say a number is a square based only on its units digit. This rule only helps elimRead more
Even though square numbers always end in 0, 1, 4, 5, 6 or 9, not all numbers ending in these digits are squares. For instance, 36 and 16 are squares ending in 6, but 26 also ends in 6 and is not a square. So, we cannot say a number is a square based only on its units digit. This rule only helps eliminate some possibilities, not confirm squareness.
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