Question: Does every number have an even number of factors?
Answer:No.
Explanation: While most factors come in “partner pairs” (like 1 and 6 for the number 6), sometimes the numbers in the pair are the same (like 2 × 2 = 4), which means they do not form two distinct factors.

Question: Can you use this insight to find more numbers with an odd number of factors?
Answer:Yes, all square numbers have an odd number of factors.
Explanation: A number that can be expressed as the product of a number with itself is a square number. They have an odd number of factors because they have one specific factor which, when multiplied by itself, equals that number.

Question: Write the locker numbers that remain open.
Answer:1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Explanation: The lockers that remain open are those toggled an odd number of times, which matches the number of factors the locker number has. Since only square numbers have an odd number of factors, the perfect squares from 1 to 100 will remain open.

Question: Which are the five lockers that were touched exactly twice (representing the passcode)?
Answer:The prime numbers: 2, 3, 5, 7, and 11.
Explanation: Lockers toggled exactly twice correspond to numbers with exactly two factors. These are prime numbers, since a prime number’s only factors are 1 and the number itself.

Question: Which of the following numbers have the digit 6 in the units place? (i) 38² (ii) 34² (iii) 46² (iv) 56² (v) 74² (vi) 82²
Answer:(ii) 34², (iii) 46², (iv) 56², and (v) 74².
Explanation: Based on the patterns of perfect squares, any number ending in 4 or 6 will have a square that ends in 6 (e.g., $4^2 = 16$, $6^2 = 36$, $14^2 = 196$).

Question: If a number contains 3 zeros at the end, how many zeros will its square have at the end?
Answer:Six zeros.
Explanation: When squaring a number, the number of zeros at the end is doubled. For example, $10^2$ has two zeros, and $100^2$ has four zeros.

Question: What do you notice about the number of zeros at the end of a number and the number of zeros at the end of its square? Will this always happen? Can we say that squares can only have an even number of zeros at the end?
Answer:The number of zeros at the end of the square is double the number of zeros at the end of a number. Yes, this will always happen. Yes, squares can only have an even number of zeros at the end.
Explanation: Squaring a number inherently doubles its trailing zeros, ensuring that a perfect square will never end with an odd number of zeros.

Question: What can you say about the parity of a number and its square?
Answer:The square of an even number is even, and the square of an odd number is odd.
Explanation: This is a consistent mathematical pattern verified in the answer key provided in the text.

Question: Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?
Answer:Yes. If $p$ and $q$ are successive perfect square numbers, then there will be $(q – p – 1)$ numbers between them.
Explanation: This formula calculates the exact number of non-square integers nestled between any two consecutive perfect square numbers.

Question: Is 156 a perfect square?
Answer:No.
Explanation: By looking at its prime factorisation ($2 \times 2 \times 3 \times 13$), we can see that these factors cannot be paired up into two equal groups, which is a requirement for a perfect square.

Question: Is 3375 a perfect cube?
Answer:Yes, it equals $15^3$.
Explanation: The prime factorisation of 3375 is $3 \times 3 \times 3 \times 5 \times 5 \times 5$. Because these factors can be grouped into three identical groups of $(3 \times 5)$, it forms a perfect cube.

Question: Is 500 a perfect cube?
Answer:No.
Explanation: The prime factorisation of 500 is $2 \times 2 \times 5 \times 5 \times 5$. The factors cannot be split into three identical groups, meaning it is not a perfect cube.

Question: Can a cube end with exactly two zeroes (00)? Explain.
Answer:No.
Explanation: While not explicitly spelled out with an equation in the text, the answer key confirms this is false. Based on the logic of multiplication, a number ending in one zero will have three zeros when cubed, meaning a perfect cube ending in zeros will have them in multiples of three.

Question: Can you tell what this sum (91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109) is without doing the calculation?
Answer:$10^3 = 1000$.
Explanation: There is a distinct pattern where consecutive perfect cubes are formed by adding sequential odd numbers. For example, $1^3 = 1$, $2^3 = 3 + 5$, and $3^3 = 7 + 9 + 11$. This sequence of ten odd numbers represents the 10th set in the pattern, equaling $10^3$.

Question: Find the cube roots of these numbers: (i) $\sqrt{64}$ (ii) $\sqrt{512}$ (iii) $\sqrt{729}$.
Answer:(i) 4, (ii) 8, (iii) 9.
Explanation: These are standard perfect cubes derived from matching the prime factors into three equal groups, verified by the text’s answer key.

“Figure it Out” Section: Squares

1. Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089.
   Answer:(i), (ii), and (iii).
   Explanation: The units digit can tell us when a number is not a square. If a number ends with 2, 3, 7, or 8, it is definitely not a square.
   2. Which one among 64², 108², 292², 36² has the last digit 4?
   Answer:108² and 292².
   Explanation: Looking at the units digits of the base numbers, $8^2 = 64$ and $2^2 = 4$. Both result in a final unit digit of 4.
   3. Given 125² = 15625, what is the value of 126²?
   Answer:(iv) 15625 + 251.
   Explanation: According to the provided answers, this relies on the mathematical pattern of adding consecutive odd numbers to find the next square.
   4. Find the length of the side of a square whose area is 441 m².
   Answer:21 m.
   Explanation: The area of a square is the product of its sides ($s \times s = s^2$). The square root of 441 is 21.
   5. Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.
   Answer:900.
   Explanation: Verified in the answer key.
   6. Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.
   Answer: Multiplied by 3; the square root of the product is 168.
   Explanation: A perfect square requires all prime factors to be in pairs. Multiplying by 3 completes the necessary pairs for 9408, resulting in a square root of 168.
   7. How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100.
   Answer:(i) 32, (ii) 198.
   Explanation: Using the formula $(q – p – 1)$ for successive squares $p$ and $q$, the calculations are $(17^2 – 16^2 – 1)$ and $(100^2 – 99^2 – 1)$.
   8. In the following pattern, fill in the missing numbers: 4² + 5² + 20² = (___)²; 9² + 10² + (___)² = (___)².
   Answer:4² + 5² + 20² = (21)²; 9² + 10² + (90)² = (91)².
   Explanation: By observing the pattern given in the text (e.g., $2^2 + 3^2 + 6^2 = 7^2$, $3^2 + 4^2 + 12^2 = 13^2$), the third number on the left is the product of the first two bases, and the base on the right is one more than that product.

“Figure it Out” Section: Cubes

1. Find the cube roots of 27000 and 10648.
   Answer:30 and 22.
   Explanation: Because $30 \times 30 \times 30 = 27000$ and $22 \times 22 \times 22 = 10648$.
   2. What number will you multiply by 1323 to make it a cube number?
   Answer:7.
   Explanation: Confirmed by the answer key to group the prime factors into identical triplets.
   3. State true or false. (i) The cube of any odd number is even. (ii) There is no perfect cube that ends with 8. (iii) The cube of a 2-digit number may be a 3-digit number. (iv) The cube of a 2-digit number may have seven or more digits. (v) Cube numbers have an odd number of factors.
   Answer:All statements are False.
   Explanation:
   (i) False: The cube of an odd number is odd (e.g., $3^3 = 27$).
   (ii) False: $2^3 = 8$.
    (iii) False: The smallest 2-digit number is 10, and $10^3 = 1000$ (a 4-digit number).
    (iv) False: Verified in the answer key.
    (v) False: It is square numbers that have an odd number of factors, not cubes.
   4. You are told that 1331 is a perfect cube. Can you guess without factoris  ation what its cube root is? Similarly, guess the cube roots of 4913, 12167, and 32768.
   Answer:Yes, it is 11. The remaining cube roots are 17, 23, and 32.
   Explanation: The text provides these directly in the answers.
   5. Which of the following is the greatest? (i) 67³ – 66³ (ii) 43³ – 42³ (iii) 67² – 66² (iv) 43² – 42².
   Answer:(i) 67³ – 66³ is the greatest.
   Explanation: The difference between consecutive cubes grows significantly faster and is much larger than the difference between consecutive squares, especially with larger base numbers.

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