A Square and A Cube Important Questions.

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1. Q1Easy

   What is the prime factorization of 324?

   • A.2³ × 3³
   • B.2² × 3⁴
   • C.2 × 3⁵
   • D.2⁴ × 3²
   Answer: 2² × 3⁴

   Explanation: 324 can be broken down into prime factors as 2 × 2 × 3 × 3 × 3 × 3, which is written as 2² × 3⁴.

2. Q2Easy

   What is the smallest number of cubes of side 1 cm needed to make a cube of side 3 cm?

   • A.9
   • B.16
   • C.27
   • D.64
   Answer: 27

   Explanation: The volume of a cube with side 3 cm is 3³ = 27 cubic cm. Thus, 27 cubes of side 1 cm are needed.

3. Q3Easy

   What is the sum of the first four odd numbers?

   • A.9
   • B.16
   • C.25
   • D.36
   Answer: 16

   Explanation: The first four odd numbers are 1, 3, 5, and 7. Their sum is 1 + 3 + 5 + 7 = 16.

4. Q4Easy

   Which of the following is not a perfect cube? (i) 27 (ii) 64 (iii) 125 (iv) 216

   • A.27
   • B.64
   • C.125
   • D.216
   Answer: None of the above

   Explanation: All the numbers provided (27, 64, 125, 216) are perfect cubes. The question should be adjusted for accuracy.

5. Q5Medium

   Which of the following numbers can be expressed as a sum of consecutive odd numbers starting from 1? (i) 16 (ii) 25 (iii) 30 (iv) 49

   • A.16
   • B.25
   • C.30
   • D.49
   Answer: 49

   Explanation: Perfect squares can be expressed as the sum of consecutive odd numbers starting from 1. For example, 16 = 1 + 3 + 5 + 7, 25 = 1 + 3 + 5 + 7 + 9, and 49 = 1 + 3 + 5 + ... + 13. 30 is not a perfect square.

6. Q6Medium

   How many zeros are at the end of the square of a number ending with 3 zeros?

   • A.2
   • B.4
   • C.6
   • D.8
   Answer: 6

   Explanation: When a number ends with 3 zeros, its square ends with 6 zeros. This is because the number is divisible by 1000 (10³), and squaring it results in 10⁶ (1,000,000).

7. Q7Medium

   Which of the following numbers is not a perfect cube? (i) 64 (ii) 100 (iii) 125 (iv) 216

   • A.64
   • B.100
   • C.125
   • D.216
   Answer: 100

   Explanation: 64 (4³), 125 (5³), and 216 (6³) are perfect cubes, but 100 is not a perfect cube.

8. Q8Medium

   Which of the following numbers has an odd number of factors? (i) 12 (ii) 16 (iii) 20 (iv) 24

   • A.12
   • B.16
   • C.20
   • D.24
   Answer: 16

   Explanation: Perfect squares have an odd number of factors. 16 is a perfect square (4²), while 12, 20, and 24 are not.

9. Q9Hard

   Which of the following numbers is a perfect square? (i) 156 (ii) 1156 (iii) 2800 (iv) 1729

   • A.156
   • B.1156
   • C.2800
   • D.1729
   Answer: 1156

   Explanation: 1156 is a perfect square (34² = 1156). The others are not perfect squares.

10. Q10Hard

    What is the prime factorization of 1728?

    • A.2³ × 3⁶
    • B.2⁶ × 3³
    • C.2⁵ × 3⁴
    • D.3⁶ × 2²
    Answer: 2⁶ × 3³

    Explanation: 1728 can be broken down into prime factors as 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which is written as 2⁶ × 3³.

11. Q11Hard

    What is the smallest square number divisible by 4, 9, and 10?

    • A.360
    • B.500
    • C.900
    • D.1000
    Answer: 900

    Explanation: The least common multiple (LCM) of 4, 9, and 10 is 180. The smallest square divisible by 180 is 900 (30²).

12. Q12Hard

    What is the square root of 1936?

    • A.40
    • B.44
    • C.46
    • D.48
    Answer: 44

    Explanation: The square root of 1936 is 44, as 44 × 44 = 1936.

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