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Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 3

Squares, Cubes and Their Roots; Playing with Numbers

What to remember

  • A perfect square ends only in 0, 1, 4, 5, 6 or 9, and the number of zeros at its end is always even; the square of n equals the sum of the first n odd numbers.
  • A cube keeps the "parity" of its number (odd gives odd, even gives even), and the units digit of a cube tells you the units digit of its cube root.
  • Prime factorisation is the master tool: pairs of equal factors give a square, triplets give a cube, and the factor list also gives HCF, LCM and the count of factors.

Squares and square roots

A square number is the product of a number with itself. Write n² for the square of n. The first squares to know by heart are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625.

Properties of squares

  • The units digit of a square is 0, 1, 4, 5, 6 or 9. A number that ends in 2, 3, 7 or 8 is never a perfect square.
  • A number ending in an odd count of zeros (one zero, three zeros) is not a perfect square.
  • The square of an even number is even and the square of an odd number is odd.
  • The square of a number ending in 1 or 9 ends in 1; ending in 4 or 6 ends in 6; ending in 5 ends in 25.
  • Between n² and (n+1)² there are 2n numbers that are not perfect squares. Example: between 12² and 13² there are 24 such numbers.
  • (n+1)² − n² = 2n + 1. So each square is the previous square plus the next odd number.
  • The sum of the first n odd numbers is n². Example: 1 + 3 + 5 + 7 + 9 = 25 = 5².
  • Pythagorean triplet: for any natural number m greater than 1, (2m, m² − 1, m² + 1) is a triplet. For m = 4 we get 8, 15, 17. For m = 5 we get 10, 24, 26.
  • A number with n digits that is a perfect square has a root with n/2 digits (n even) or (n+1)/2 digits (n odd).

Short methods

  • Squaring a number ending in 5: take the tens part a, multiply a by (a+1), and write 25 after it. 35²: 3 × 4 = 12, so 1225. 85²: 8 × 9 = 72, so 7225.
  • Squaring numbers near 100: 99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
  • Identities: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b).

Finding a square root

  • 1. Prime factorisation: write the factors in pairs and take one from each pair. 1764 = 2² × 3² × 7², so √1764 = 2 × 3 × 7 = 42.
  • 2. Long division: make pairs of digits from the right (for decimals, from the decimal point in both directions) and divide step by step. This works for numbers that are not easy to factorise.
  • 3. Estimation: if a number is between two known squares, its root is between their roots. 500 lies between 22² = 484 and 23² = 529, so √500 is between 22 and 23.
  • Square root of a fraction: √(a/b) = √a / √b. √(196/361) = 14/19.
  • Square root of a decimal: the number of decimal places in the root is half those in the number. √0.0169 = 0.13.
  • Useful approximate values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236.

Making a perfect square

Factorise, find the primes whose power is odd, and multiply (or divide) by those primes.

  • 72 = 2³ × 3². The prime 2 is unpaired, so multiply by 2 to get 144 = 12².
  • To find the least square divisible by 6, 8 and 12, take LCM = 24 = 2³ × 3, then multiply by 2 × 3 = 6 to get 144.

Cubes and cube roots

A cube is n × n × n. The first cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331 (11³), 1728 (12³).

Properties

  • The cube of an odd number is odd; the cube of an even number is even.
  • The cube of a negative number is negative: (−4)³ = −64, so ∛(−343) = −7.
  • A number with an unpaired triplet is not a perfect cube. Prime factors of a perfect cube occur in groups of three.
  • The sum of the cubes of the first n natural numbers equals (n(n+1)/2)². Example: 1³ + 2³ + 3³ + 4³ = 100 = 10².
  • (a + b)³ = a³ + 3a²b + 3ab² + b³. a³ + b³ = (a + b)(a² − ab + b²). a³ − b³ = (a − b)(a² + ab + b²).

Units digits of cubes

Number ends in0123456789
Cube ends in0187456329

Notice that only 2 ↔ 8 and 3 ↔ 7 swap; the rest repeat.

Cube root by prime factorisation: group factors in triplets. 2744 = 2³ × 7³, so ∛2744 = 14.

Cube root by estimation (for perfect cubes up to 6 digits): Take the last three digits to get the units digit of the root using the table. Take the remaining part and find the largest cube below it for the tens digit. For 17576: last digit 6 gives units digit 6; the remaining part is 17, between 8 (2³) and 27 (3³), so the tens digit is 2. Answer: 26.

Making a perfect cube: 108 = 2² × 3³. The prime 2 appears twice, so multiply by 2 to get 216 = 6³. For 1250 = 2 × 5⁴, divide by 2 × 5 = 10 to get 125.

For decimals, the root has one decimal place for every three places in the number: ∛0.001728 = 0.12.

Playing with numbers

Divisibility rules

ByRule
2Last digit is 0, 2, 4, 6 or 8
3Sum of digits is divisible by 3
4Last two digits divisible by 4
5Last digit is 0 or 5
6Divisible by both 2 and 3
8Last three digits divisible by 8
9Sum of digits divisible by 9
10Last digit is 0
11Difference between sum of digits in odd places and sum in even places is 0 or a multiple of 11

Examples: 2728: (2 + 2) − (7 + 8) = −11, so it is divisible by 11. 52x4 is divisible by 9 when 5 + 2 + x + 4 = 11 + x is a multiple of 9, so x = 7. 7512 is divisible by 8 because 512 = 8 × 64.

Primes and composites

  • A prime has exactly two factors, 1 and itself. 2 is the only even prime. 1 is neither prime nor composite.
  • There are 8 primes below 20 (2, 3, 5, 7, 11, 13, 17, 19) and 25 primes below 100.
  • Twin primes differ by 2: (3, 5), (11, 13), (17, 19), (41, 43).
  • Co-prime numbers have HCF 1. They need not be prime themselves: 21 and 22 are co-prime.
  • Perfect numbers equal the sum of their proper factors: 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14.
  • The Sieve of Eratosthenes finds primes by crossing out multiples. It is a good classroom activity on a hundreds chart.

Factors, HCF and LCM

  • If N = p^a × q^b, the number of factors is (a + 1)(b + 1). 36 = 2² × 3² has 9 factors; 72 = 2³ × 3² has 12 factors.
  • HCF is the product of the lowest powers of common primes; LCM is the product of the highest powers of all primes.
  • For two numbers: HCF × LCM = product of the numbers. If the product is 2160 and the HCF is 12, the LCM is 180.
  • HCF of 36 and 48 is 12; LCM is 144.

Number curiosities

  • 1729 is the Hardy–Ramanujan number: the smallest number that is the sum of two cubes in two ways (1³ + 12³ = 9³ + 10³).
  • Sum of first n natural numbers = n(n+1)/2; first n even numbers = n(n+1); first n odd numbers = n².
  • A palindrome reads the same both ways (1331, 2552).

Classroom angle: Use square tiles and dot patterns to show that n² is a square arrangement and that adding an L-shaped border of 2n + 1 tiles makes the next square. A 1 cm cube model of side 3 shows 27 unit cubes.

Comparison table

FeatureSquareCube
Grouping of prime factorsPairsTriplets
Possible units digits0, 1, 4, 5, 6, 9Any digit 0 to 9
Negative base givesPositiveNegative
Root of a negative numberNot a real numberReal, negative
Zeros at the endEven countMultiple of 3
Geometric meaningArea of a squareVolume of a cube

Exam traps

  • 1 is not a prime number, and 2 is the only even prime.
  • A number ending in 5 is divisible by 5 but its square ends in 25, not 5 only.
  • A perfect square cannot end in an odd number of zeros, but a number like 1600 (two zeros) can be square.
  • Between n² and (n+1)² there are 2n non-squares, not 2n + 1 or n.
  • A number divisible by both 4 and 6 need not be divisible by 24; it must be divisible by the LCM, 12.
  • Co-prime does not mean prime: 8 and 9 are co-prime.
  • The unit digit of a cube of a number ending in 2 is 8, not 4; for 3 it is 7, not 9.
  • Divisibility by 9 needs the sum of all digits, not just the last digit.

One-liners

  • √1764 = 42 and ∛2744 = 14.
  • 1 + 3 + 5 + 7 + 9 + 11 = 36 = 6².
  • 35² = 1225 and 99² = 9801.
  • The smallest number to multiply 72 by for a square is 2.
  • The smallest number to multiply 108 by for a cube is 2.
  • 17576 is the cube of 26.
  • 28 is a perfect number.
  • 1729 is the Hardy–Ramanujan (taxicab) number.
  • The number of factors of 72 is 12.
  • The Pythagorean triplet for m = 5 is 10, 24, 26.
  • 1 + 2 + 3 + ... + 20 = 210.
  • 101 is the smallest three-digit prime.

Practice questions

  1. What is the square root of 1764?

    1. 48
    2. 44
    3. 46
    4. 42
    Answer

    D. 42

    1764 = 2² × 3² × 7², so the root is 2 × 3 × 7 = 42.

  2. What is the cube root of 2744?

    1. 12
    2. 16
    3. 14
    4. 18
    Answer

    C. 14

    2744 = 2³ × 7³, so the cube root is 2 × 7 = 14.

  3. Which of these numbers is NOT a perfect square?

    1. 1450
    2. 1296
    3. 1156
    4. 1369
    Answer

    A. 1450

    1450 ends in one zero (an odd count), so it cannot be a square. 1156 = 34², 1296 = 36², 1369 = 37².

  4. What is the sum of the first 15 odd natural numbers?

    1. 255
    2. 235
    3. 215
    4. 225
    Answer

    D. 225

    The sum of the first n odd numbers is n², so 15² = 225.

  5. How many natural numbers lie between 12² and 13² (both excluded)?

    1. 23
    2. 25
    3. 24
    4. 12
    Answer

    C. 24

    Between n² and (n+1)² there are 2n numbers, so 2 × 12 = 24.

  6. Using the triplet (2m, m² − 1, m² + 1) with m = 5, the Pythagorean triplet is

    1. 12, 24, 26
    2. 10, 24, 26
    3. 10, 24, 25
    4. 10, 25, 26
    Answer

    B. 10, 24, 26

    2m = 10, m² − 1 = 24, m² + 1 = 26. Check: 100 + 576 = 676 = 26².

  7. What is the units digit of 27³?

    1. 7
    2. 9
    3. 3
    4. 1
    Answer

    C. 3

    The cube of a number ending in 7 ends in 3 (7³ = 343).

  8. What is the smallest number by which 72 must be multiplied to get a perfect square?

    1. 2
    2. 3
    3. 6
    4. 8
    Answer

    A. 2

    72 = 2³ × 3². The factor 2 is unpaired, so multiply by 2 to get 144 = 12².

  9. What is the smallest number by which 108 must be multiplied to get a perfect cube?

    1. 3
    2. 2
    3. 4
    4. 6
    Answer

    B. 2

    108 = 2² × 3³. Multiplying by 2 gives 2³ × 3³ = 216 = 6³.

  10. What is the smallest number by which 1250 must be divided to get a perfect cube?

    1. 2
    2. 5
    3. 25
    4. 10
    Answer

    D. 10

    1250 = 2 × 5⁴. Dividing by 2 × 5 = 10 leaves 125 = 5³.

  11. What is the square root of 0.0169?

    1. 0.13
    2. 1.3
    3. 0.17
    4. 0.013
    Answer

    A. 0.13

    0.0169 = 169/10000, and √169 = 13, √10000 = 100, so the root is 0.13.

  12. What is the square root of 196/361?

    1. 13/19
    2. 16/19
    3. 14/17
    4. 14/19
    Answer

    D. 14/19

    √196 = 14 and √361 = 19.

  13. What is the smallest perfect square that is divisible by 6, 8 and 12?

    1. 576
    2. 64
    3. 144
    4. 36
    Answer

    C. 144

    LCM = 24 = 2³ × 3. Multiply by 2 × 3 = 6 to pair the factors: 144 = 12².

  14. What is the value of 1³ + 2³ + 3³ + 4³ + 5³?

    1. 325
    2. 225
    3. 200
    4. 125
    Answer

    B. 225

    The sum equals (5 × 6 / 2)² = 15² = 225.

  15. What is the cube root of 0.001728?

    1. 1.2
    2. 0.012
    3. 0.12
    4. 0.14
    Answer

    C. 0.12

    0.12 × 0.12 = 0.0144 and 0.0144 × 0.12 = 0.001728.

  16. A perfect-square number has 6 digits. How many digits does its square root have?

    1. 2
    2. 4
    3. 6
    4. 3
    Answer

    D. 3

    For an even count n of digits, the root has n/2 digits, so 6/2 = 3.

  17. What is the value of 35²?

    1. 1225
    2. 1125
    3. 1325
    4. 1235
    Answer

    A. 1225

    For a number ending in 5: 3 × 4 = 12, then write 25, giving 1225.

  18. What is the value of 99²?

    1. 9701
    2. 9801
    3. 9899
    4. 9901
    Answer

    B. 9801

    (100 − 1)² = 10000 − 200 + 1 = 9801.

  19. The number 1729, known as the Hardy–Ramanujan number, is the smallest number that can be written as

    1. the sum of two cubes in two different ways
    2. the sum of its proper factors
    3. a perfect square that is also a cube
    4. a prime number of four digits
    Answer

    A. the sum of two cubes in two different ways

    1729 = 1³ + 12³ = 9³ + 10³.

  20. Which is the smallest three-digit prime number?

    1. 111
    2. 107
    3. 101
    4. 103
    Answer

    C. 101

    100 is even and 101 has no factor up to 10 (not divisible by 2, 3, 5, 7), so 101 is prime.

  21. How many prime numbers are there between 1 and 20?

    1. 7
    2. 8
    3. 9
    4. 10
    Answer

    B. 8

    They are 2, 3, 5, 7, 11, 13, 17, 19.

  22. Which of these numbers is divisible by 11?

    1. 4523
    2. 5612
    3. 3147
    4. 2728
    Answer

    D. 2728

    For 2728: (2 + 2) − (7 + 8) = −11, a multiple of 11.

  23. If the number 52x4 is divisible by 9, what is the digit x?

    1. 7
    2. 2
    3. 5
    4. 9
    Answer

    A. 7

    Digit sum 5 + 2 + x + 4 = 11 + x must be a multiple of 9, so x = 7 (sum 18).

  24. Which of these numbers is divisible by 8?

    1. 7514
    2. 7512
    3. 7516
    4. 7518
    Answer

    B. 7512

    The last three digits 512 = 8 × 64. The others end in 514, 516, 518, none of which is a multiple of 8.

  25. The product of two numbers is 2160 and their HCF is 12. What is their LCM?

    1. 144
    2. 240
    3. 180
    4. 360
    Answer

    C. 180

    HCF × LCM = product, so LCM = 2160 / 12 = 180.

  26. How many factors does 36 have?

    1. 12
    2. 6
    3. 8
    4. 9
    Answer

    D. 9

    36 = 2² × 3², so (2 + 1)(2 + 1) = 9 factors.

  27. How many factors does 72 have?

    1. 10
    2. 12
    3. 18
    4. 9
    Answer

    B. 12

    72 = 2³ × 3², so (3 + 1)(2 + 1) = 12 factors.

  28. Which is the only even prime number?

    1. 1
    2. 4
    3. 6
    4. 2
    Answer

    D. 2

    2 has only factors 1 and 2; every other even number has 2 as an extra factor.

  29. Which of these is a perfect number?

    1. 28
    2. 30
    3. 36
    4. 24
    Answer

    A. 28

    Proper factors of 28: 1 + 2 + 4 + 7 + 14 = 28.

  30. Which of the following pairs is a pair of twin primes?

    1. 19 and 23
    2. 9 and 11
    3. 17 and 19
    4. 13 and 17
    Answer

    C. 17 and 19

    Twin primes are primes that differ by 2. 17 and 19 are both prime.

  31. Which of the following is a pair of co-prime numbers?

    1. 21 and 22
    2. 15 and 25
    3. 18 and 27
    4. 14 and 21
    Answer

    A. 21 and 22

    HCF of 21 and 22 is 1. The others share the factors 7, 5 and 9 respectively.

  32. Between which two consecutive whole numbers does √500 lie?

    1. 21 and 22
    2. 24 and 25
    3. 23 and 24
    4. 22 and 23
    Answer

    D. 22 and 23

    22² = 484 and 23² = 529, and 484 < 500 < 529.

  33. What is the cube root of 17576?

    1. 28
    2. 26
    3. 34
    4. 24
    Answer

    B. 26

    Units digit 6 gives 6; the part 17 lies between 2³ and 3³, so tens digit 2. Check: 26³ = 17576.

  34. What is the value of ∛(−343)?

    1. 7
    2. −14
    3. −7
    4. −49
    Answer

    C. −7

    (−7) × (−7) × (−7) = −343.

  35. Consider these statements. 1. The cube of an odd number is odd. 2. The cube of a negative number is positive. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    A. 1 only

    A negative number cubed stays negative, so statement 2 is wrong. Statement 1 is right.

  36. Consider these statements. 1. A perfect square can end in the digit 7. 2. The square of a number ending in 5 ends in 25. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    No square ends in 2, 3, 7 or 8, so 1 is wrong. 5² = 25, 15² = 225, so 2 is right.

  37. Consider these statements. 1. A perfect square has an even number of zeros at the end. 2. The square of an odd number is odd. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    C. Both 1 and 2

    Both are standard properties of squares.

  38. Consider these statements. 1. Every prime number is odd. 2. The number 1 is a prime number. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    D. Neither 1 nor 2

    2 is an even prime, and 1 has only one factor so it is not prime.

  39. Consider these statements. 1. The sum of two odd primes is odd. 2. The cube of an even number is even. Which is/are correct?

    1. 1 only
    2. 2 only
    3. Both 1 and 2
    4. Neither 1 nor 2
    Answer

    B. 2 only

    Odd + odd = even, so 1 is wrong. Even × even × even is even, so 2 is right.

  40. Match the items: (1) Perfect number (2) Twin primes (3) Taxicab number (4) Smallest composite number with (a) 1729 (b) 4 (c) 28 (d) 5 and 7. Choose the correct matching.

    1. 1-a, 2-d, 3-c, 4-b
    2. 1-c, 2-b, 3-a, 4-d
    3. 1-d, 2-c, 3-b, 4-a
    4. 1-c, 2-d, 3-a, 4-b
    Answer

    D. 1-c, 2-d, 3-a, 4-b

    28 is perfect; 5 and 7 differ by 2; 1729 is the taxicab number; 4 is the smallest composite.

  41. A teacher arranges 169 students in a square formation with equal rows and columns. How many students stand in each row?

    1. 13
    2. 14
    3. 12
    4. 16
    Answer

    A. 13

    √169 = 13.

  42. The area of a square garden is 1296 square metres. What is its perimeter?

    1. 108 m
    2. 36 m
    3. 144 m
    4. 72 m
    Answer

    C. 144 m

    Side = √1296 = 36 m, so the perimeter is 4 × 36 = 144 m.

  43. A cube has a volume of 3375 cubic centimetres. What is the length of its edge?

    1. 25 cm
    2. 13 cm
    3. 17 cm
    4. 15 cm
    Answer

    D. 15 cm

    ∛3375 = 15, because 15³ = 3375.

  44. What is the least number that must be added to 1000 to make it a perfect square?

    1. 31
    2. 36
    3. 24
    4. 21
    Answer

    C. 24

    31² = 961 and 32² = 1024, so 1024 − 1000 = 24.

  45. What is the least number that must be subtracted from 2000 to make it a perfect square?

    1. 64
    2. 56
    3. 44
    4. 84
    Answer

    A. 64

    44² = 1936 and 45² = 2025, so 2000 − 1936 = 64.

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