A Square and A Cube — Practice

Q1. In Queen Ratnamanjuri's puzzle only certain lockers remain open at the end. These lockers are exactly the ones whose numbers have

• A. an odd number of factors
• B. an even number of factors
• C. exactly two factors
• D. exactly four factors

Q2. The locker numbers that remain open at the end of the puzzle. These are

• A. 1, 2, 3, 5, 7, 11, 13, 17, 19, 23
• B. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
• C. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
• D. 2, 4, 8, 16, 32, 64, 100, 200, 400, 800

Q3. Priya writes 7 × 7 in short form as 7². 7² is read as

• A. seven plus two
• B. seven by two
• C. seven squared
• D. seven cubed

Q4. Why numbers 1, 4, 9, 16, … are called squares: each equals the area of a square whose sidelength is a natural number. The area of a square of sidelength 5 units is

• A. 10 sq units
• B. 20 sq units
• C. 15 sq units
• D. 25 sq units

Q5. The units digits of the first 30 perfect squares. The possible units digits of any perfect square are

• A. 0, 1, 4, 5, 6 or 9
• B. 0, 2, 4, 6 or 8
• C. 1, 3, 5, 7 or 9
• D. 2, 3, 7 or 8

Q6. Aarti picks four numbers — 23, 36, 47 and 58. Using rule, which of these can be definitely said to not be a perfect square?

• A. 36
• B. 23, 47 and 58
• C. 23 and 36
• D. 36 and 58

Q7. Asks: if a number ends with 3 zeros, how many zeros will its square end with?

• A. 3 zeros
• B. 4 zeros
• C. 6 zeros
• D. 9 zeros

Q8. 1 + 3 + 5 + 7 + 9 = 25 = 5². In general, the sum of the first n odd numbers starting from 1 equals

• A. 2n
• B. 2n − 1
• C. n + 1
• D. n²

Q9. The formula for the nth odd number. Using this, the 36th odd number is

• A. 71
• B. 72
• C. 73
• D. 36

Q10. Ravi knows the area of a square is 49 sq cm. Using, the length of one side of the square is

• A. 5 cm
• B. 7 cm
• C. 9 cm
• D. 14 cm

Q11. Writes 324 = 2 × 2 × 3 × 3 × 3 × 3 and pairs the prime factors. The square root of 324 is

• A. 12
• B. 16
• C. 18
• D. 24

Q12. The numbers 1, 8, 27, … are called perfect cubes because each is obtained by

• A. multiplying a number by itself three times
• B. adding a number to itself three times
• C. multiplying a number by 3
• D. raising 3 to that number

Q13. Manju makes a cube with edge length 4 units by stacking 1-unit cubes. The total number of 1-unit cubes she needs is

• A. 12
• B. 16
• C. 48
• D. 64

Q14. The cube root is defined as the inverse operation of cubing. The cube root of 27 is

• A. 2
• B. 3
• C. 9
• D. 27

Q15. Tells the story of Hardy's taxi-cab visit to Ramanujan. The number 1729 is famous because it is the smallest number that can be expressed as

• A. the product of two cubes
• B. the sum of two cubes in two different ways
• C. the cube of a prime number
• D. the sum of three consecutive cubes