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

Numbers

What to remember

  • Place value = face value x position value; the Indian system groups digits as 3, 2, 2, ... from the right, the international system groups them in 3s.
  • Divisibility rules give quick tests: a number is divisible by 3 (or 9) if its digit sum is divisible by 3 (or 9); by 11 if the difference of the sums of alternate digits is 0 or a multiple of 11.
  • HCF x LCM = product of the two numbers. HCF divides every given number; LCM is divisible by every given number.

Numeration systems

A numeration system is a way of writing numbers with symbols.

Hindu-Arabic (decimal) system: uses ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and a place-value idea with base 10. The zero works as a place holder. This is the system used in schools today.

Roman system: uses letters. It has no zero and no place value.

SymbolIVXLCDM
Value1510501005001000

Rules: a symbol repeated is added (III = 3), but I, X, C, M are not repeated more than three times; V, L, D are never repeated. A smaller symbol before a larger one is subtracted (IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900); a smaller symbol after a larger one is added (VI = 6). Only I, X and C are subtracted, and I is subtracted only from V and X; X only from L and C; C only from D and M. Example: 1994 = MCMXCIV; XLVIII = 48.

Other bases: the binary system (base 2) uses 0 and 1. A number such as 1011 in binary = 1x8 + 0x4 + 1x2 + 1x1 = 11 in decimal. Computers use binary.

Place value and face value

  • Face value: the value of the digit itself, whatever its place.
  • Place value: the value of the digit by its position.

In 7,48,305: the face value of 4 is 4; its place value is 4 x 10,000 = 40,000.

Indian system: ones, tens, hundreds, thousands, ten thousands, lakhs, ten lakhs, crores, ten crores. Commas come after the first three digits from the right and then after every two digits: 12,34,56,789.

International system: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions. Commas come after every three digits: 123,456,789.

IndianInternational
1 lakh = 100,000100 thousand
10 lakh = 1,000,0001 million
1 crore = 10,000,00010 million

Expanded form: 5,306 = 5 x 1000 + 3 x 100 + 0 x 10 + 6 x 1.

Number of digits: the smallest n-digit number is 10^(n-1) and the largest is 10^n - 1. The largest 4-digit number is 9999 and the smallest is 1000.

Kinds of numbers

  • Natural numbers (N): 1, 2, 3, ... The smallest is 1.
  • Whole numbers (W): 0, 1, 2, 3, ... The smallest is 0.
  • Integers (Z): ..., -2, -1, 0, 1, 2, ...
  • Even numbers: divisible by 2. Odd numbers: not divisible by 2.
  • Prime number: has exactly two factors, 1 and itself (2, 3, 5, 7, 11, 13). The number 2 is the only even prime. The number 1 is neither prime nor composite.
  • Composite number: has more than two factors (4, 6, 8, 9, 10).
  • Co-prime (relatively prime) numbers: two numbers whose HCF is 1 (8 and 15). They need not be prime themselves.
  • Twin primes: primes that differ by 2 (3 and 5, 11 and 13, 17 and 19).
  • Perfect number: equals the sum of its proper factors (6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14).
  • Perfect square: square of a whole number (1, 4, 9, 16, 25).

Sieve of Eratosthenes: a method to find primes. List numbers, cross out multiples of 2, then 3, then 5, and so on. The numbers left are primes. Teachers use this activity in class.

Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. There are 15 of them.

Factors and multiples

  • A factor of a number divides it exactly. Factors of 12: 1, 2, 3, 4, 6, 12.
  • A multiple of a number is the product of the number with any natural number. Multiples of 4: 4, 8, 12, ...
  • 1 is a factor of every number. Every number is a factor and a multiple of itself.
  • Every factor is less than or equal to the number; every multiple is greater than or equal to the number.
  • The number of factors of a number is finite; the number of multiples is infinite.

Prime factorisation: writing a number as a product of primes. 360 = 2 x 2 x 2 x 3 x 3 x 5 = 2^3 x 3^2 x 5. The Fundamental Theorem of Arithmetic says that every composite number has a unique prime factorisation, apart from the order of the factors.

Number of factors: if N = a^p x b^q x c^r (a, b, c primes), the number of factors is (p+1)(q+1)(r+1). For 360: (3+1)(2+1)(1+1) = 24.

Divisibility rules

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

Example: 2728 is divisible by 11 because (2 + 2) - (7 + 8) = 4 - 15 = -11. Example: 5346: digit sum 18, so it is divisible by 9 and 3; last digit even, so by 2 and 6.

If a number is divisible by two co-prime numbers, it is divisible by their product (for example 12 = 3 x 4). Divisibility by 12 needs both 3 and 4; by 15 needs both 3 and 5.

HCF (GCD)

The Highest Common Factor (or Greatest Common Divisor) of given numbers is the largest number that divides each of them exactly.

Methods:

  • 1. Prime factorisation: take the common primes with the smallest powers.
  • 2. Long division (Euclid's algorithm): divide the larger by the smaller, then the divisor by the remainder, and so on. The last non-zero divisor is the HCF.

Example: HCF of 36 and 48. 36 = 2^2 x 3^2; 48 = 2^4 x 3. HCF = 2^2 x 3 = 12.

Example by division: HCF of 252 and 105: 252 = 2 x 105 + 42; 105 = 2 x 42 + 21; 42 = 2 x 21 + 0. HCF = 21.

LCM

The Lowest Common Multiple of given numbers is the smallest number that is divisible by each of them.

Methods: prime factorisation (take every prime with the highest power), common division, listing multiples.

Example: LCM of 12 and 18. 12 = 2^2 x 3; 18 = 2 x 3^2. LCM = 2^2 x 3^2 = 36.

Important results

ResultStatement
Product ruleFor two numbers a and b, HCF x LCM = a x b
Co-prime numbersHCF = 1, so LCM = product
HCF of fractionsHCF of numerators / LCM of denominators
LCM of fractionsLCM of numerators / HCF of denominators
DivisibilityHCF divides LCM
If one number divides the otherHCF = smaller number; LCM = larger number
Largest number dividing a, b, c leaving the same remainder rHCF of (a - r), (b - r), (c - r)
Smallest number divisible by a, b, cLCM of a, b, c
Smallest number which leaves remainder r on division by a, b, cLCM + r

Example: HCF of two numbers is 6 and their LCM is 72. If one is 18, the other is 6 x 72 / 18 = 24.

Example: Least number divisible by 12, 15 and 20: 12 = 2^2 x 3; 15 = 3 x 5; 20 = 2^2 x 5. LCM = 2^2 x 3 x 5 = 60.

Example: Largest number that divides 43 and 91 and leaves remainder 3 each time: HCF of 40 and 88 = 8.

Word problems: when to use HCF and LCM

  • HCF: cutting the longest piece, making the largest equal groups, the greatest capacity of a measure that fills containers exactly, the largest square tile for a floor.
  • LCM: events that repeat together again (bells ringing), the least length that can be measured by different rods, the smallest quantity that can be divided into the given sizes.

Example: Three bells toll at intervals of 6, 8 and 12 minutes. They toll together at 9:00. Next together after LCM(6, 8, 12) = 24 minutes, at 9:24.

Example: Largest tile for a floor 520 cm x 440 cm: HCF(520, 440) = 40 cm. (520 = 440 + 80; 440 = 5 x 80 + 40; 80 = 2 x 40.)

Other useful results

  • Sum of first n natural numbers = n(n+1)/2. Sum of first n odd numbers = n^2. Sum of first n even numbers = n(n+1).
  • A number is a perfect square if every prime in its factorisation has an even power.
  • Square numbers end only in 0, 1, 4, 5, 6 or 9.
  • Divisibility of the sum or difference: if a number divides two numbers, it divides their sum and difference.

Teaching numbers in the classroom

  • Use concrete objects (beads, sticks, counters), then pictures, then symbols (concrete-pictorial-abstract).
  • Use the abacus and place value charts with ones, tens and hundreds to build place value.
  • Group objects in bundles of ten to teach tens and hundreds.
  • Teach divisibility by discovery: let children find the patterns in the multiplication table.
  • Teach factors by arranging counters in rectangles (array method); this also shows primes (only one array).
  • Use games such as the sieve, number charts and number lines.
  • Common errors: confusing face value and place value; treating 1 as prime; writing 4 as IIII in Roman numerals under the standard rules; mixing HCF and LCM in word problems.

Exam traps

  • 1 is neither prime nor composite; 2 is the only even prime.
  • Co-prime numbers need not be prime (8 and 9).
  • Face value never changes; place value changes with position.
  • HCF is never larger than the smallest number; LCM is never smaller than the largest number.
  • The Roman numeral V, L and D are never repeated and never subtracted.
  • Divisibility by 6 needs both 2 and 3; divisibility by 8 uses the last three digits, not two.
  • Zero is a whole number but not a natural number.
  • The rule for 11 uses the difference of alternate digit sums, not the sum of all digits.

One-liners

  • 1. The Hindu-Arabic system has ten digits and base 10.
  • 2. In the Indian system, 1 crore = 100 lakh.
  • 3. The smallest natural number is 1; the smallest whole number is 0.
  • 4. The only even prime is 2.
  • 5. 1 is neither prime nor composite.
  • 6. A perfect number such as 6 equals the sum of its proper factors.
  • 7. A number divisible by 9 has a digit sum divisible by 9.
  • 8. HCF x LCM = product of two numbers.
  • 9. The largest 4-digit number is 9999.
  • 10. 360 has 24 factors.
  • 11. The sieve of Eratosthenes finds primes.
  • 12. The Roman numeral for 49 is XLIX.

Practice questions

  1. The Roman numeral for 49 is

    1. IL
    2. XIL
    3. XLIX
    4. XXXXIX
    Answer

    C. XLIX

    49 = 40 + 9 = XL + IX = XLIX; I cannot be subtracted from L.

  2. The place value of 7 in 3,72,418 is

    1. 7
    2. 700,000
    3. 7,000
    4. 70,000
    Answer

    D. 70,000

    The 7 is in the ten-thousands place: 7 x 10,000 = 70,000.

  3. The face value of 5 in 4,50,312 is

    1. 500
    2. 5
    3. 50,000
    4. 5,00,000
    Answer

    B. 5

    Face value is the digit itself.

  4. In the Indian system, 1 crore equals

    1. 100 lakh
    2. 1000 lakh
    3. 10,000 lakh
    4. 10 lakh
    Answer

    A. 100 lakh

    1 crore = 1,00,00,000 = 100 lakh.

  5. The smallest prime number is

    1. 1
    2. 3
    3. 0
    4. 2
    Answer

    D. 2

    2 is the smallest prime and the only even prime.

  6. Which number is divisible by 9?

    1. 4567
    2. 7254
    3. 8231
    4. 5123
    Answer

    B. 7254

    7+2+5+4 = 18, divisible by 9; the other digit sums are 11, 22 and 14.

  7. Which number is divisible by 11?

    1. 2728
    2. 2731
    3. 2729
    4. 2727
    Answer

    A. 2728

    (2 + 2) - (7 + 8) = -11, a multiple of 11.

  8. The number of factors of 360 is

    1. 20
    2. 18
    3. 24
    4. 30
    Answer

    C. 24

    360 = 2^3 x 3^2 x 5, so (3+1)(2+1)(1+1) = 24.

  9. The HCF of 36 and 48 is

    1. 6
    2. 144
    3. 12
    4. 24
    Answer

    C. 12

    36 = 2^2 x 3^2 and 48 = 2^4 x 3; common part 2^2 x 3 = 12.

  10. The LCM of 12 and 18 is

    1. 36
    2. 6
    3. 216
    4. 72
    Answer

    A. 36

    12 = 2^2 x 3 and 18 = 2 x 3^2; LCM = 2^2 x 3^2 = 36.

  11. The HCF of two numbers is 6 and their LCM is 72. If one number is 18, the other is

    1. 48
    2. 24
    3. 12
    4. 36
    Answer

    B. 24

    Other = (6 x 72) / 18 = 24.

  12. The LCM of 12, 15 and 20 is

    1. 180
    2. 30
    3. 120
    4. 60
    Answer

    D. 60

    12 = 2^2 x 3; 15 = 3 x 5; 20 = 2^2 x 5; LCM = 2^2 x 3 x 5 = 60.

  13. The HCF of 252 and 105 is

    1. 42
    2. 63
    3. 21
    4. 7
    Answer

    C. 21

    252 = 2 x 105 + 42; 105 = 2 x 42 + 21; 42 = 2 x 21; HCF = 21.

  14. Three bells ring every 6, 8 and 12 minutes. After how many minutes will they ring together again?

    1. 24
    2. 36
    3. 48
    4. 12
    Answer

    A. 24

    LCM of 6, 8, 12 = 24.

  15. The largest square tile that can exactly cover a floor 520 cm by 440 cm has side

    1. 10 cm
    2. 80 cm
    3. 20 cm
    4. 40 cm
    Answer

    D. 40 cm

    HCF of 520 and 440 = 40 (520 = 440 + 80; 440 = 5 x 80 + 40; 80 = 2 x 40).

  16. The smallest number exactly divisible by 8, 12 and 18 is

    1. 36
    2. 72
    3. 24
    4. 144
    Answer

    B. 72

    8 = 2^3; 12 = 2^2 x 3; 18 = 2 x 3^2; LCM = 2^3 x 3^2 = 72.

  17. The smallest number that leaves remainder 3 when divided by 6, 8 and 12 is

    1. 27
    2. 51
    3. 24
    4. 15
    Answer

    A. 27

    LCM of 6, 8, 12 is 24; 24 + 3 = 27.

  18. The largest number that divides 43 and 91 leaving remainder 3 each time is

    1. 4
    2. 16
    3. 8
    4. 12
    Answer

    C. 8

    HCF of (43 - 3) and (91 - 3) = HCF of 40 and 88 = 8.

  19. How many prime numbers are there between 1 and 50?

    1. 16
    2. 14
    3. 12
    4. 15
    Answer

    D. 15

    The primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

  20. Which pair of numbers is co-prime?

    1. 14 and 21
    2. 8 and 15
    3. 6 and 9
    4. 12 and 18
    Answer

    B. 8 and 15

    HCF of 8 and 15 is 1; the other pairs have common factors 3, 6 and 7.

  21. Which pair is a pair of twin primes?

    1. 11 and 13
    2. 9 and 11
    3. 7 and 11
    4. 13 and 17
    Answer

    A. 11 and 13

    Twin primes differ by 2; 11 and 13 are both prime.

  22. Which of the following is a perfect number?

    1. 24
    2. 30
    3. 12
    4. 28
    Answer

    D. 28

    28 = 1 + 2 + 4 + 7 + 14.

  23. The largest 3-digit number divisible by 12 is

    1. 992
    2. 999
    3. 996
    4. 984
    Answer

    C. 996

    999 / 12 gives remainder 3, so 999 - 3 = 996.

  24. The sum of the first 20 odd numbers is

    1. 210
    2. 400
    3. 200
    4. 420
    Answer

    B. 400

    The sum of the first n odd numbers is n^2, so 20^2 = 400.

  25. The binary number 1011 equals which decimal number?

    1. 10
    2. 9
    3. 13
    4. 11
    Answer

    D. 11

    1x8 + 0x4 + 1x2 + 1x1 = 11.

  26. The number of factors of 72 is

    1. 8
    2. 12
    3. 9
    4. 10
    Answer

    B. 12

    72 = 2^3 x 3^2; (3+1)(2+1) = 12.

  27. The HCF of 24, 36 and 60 is

    1. 6
    2. 12
    3. 24
    4. 4
    Answer

    B. 12

    24 = 2^3 x 3; 36 = 2^2 x 3^2; 60 = 2^2 x 3 x 5; common part 2^2 x 3 = 12.

  28. The LCM of 15, 20 and 25 is

    1. 150
    2. 600
    3. 300
    4. 100
    Answer

    C. 300

    15 = 3 x 5; 20 = 2^2 x 5; 25 = 5^2; LCM = 2^2 x 3 x 5^2 = 300.

  29. Which number is divisible by 4?

    1. 3512
    2. 3514
    3. 3518
    4. 3522
    Answer

    A. 3512

    The last two digits 12 are divisible by 4; 14, 18 and 22 are not.

  30. A teacher uses bundles of ten sticks and loose sticks to show 3 tens and 4 ones. The idea being taught is

    1. Place value
    2. Roman numerals
    3. Divisibility
    4. Prime numbers
    Answer

    A. Place value

    Bundles of ten make the tens place concrete.

  31. The Sieve of Eratosthenes is used to find

    1. HCF
    2. Perfect squares
    3. Prime numbers
    4. Roman numerals
    Answer

    C. Prime numbers

    Multiples of each number are crossed out and the primes are left.

  32. A child says 1 is a prime number. The best correction is that

    1. 1 has only one factor, but a prime has exactly two
    2. 1 is even
    3. 1 has no factors
    4. 1 is composite
    Answer

    A. 1 has only one factor, but a prime has exactly two

    A prime must have exactly two factors; 1 has one.

  33. Consider the statements about primes. 1. 1 is a prime number. 2. 2 is the only even prime number.

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

    B. 2 only

    1 is neither prime nor composite.

  34. Consider the statements about HCF and LCM. 1. The LCM of two numbers is always less than their HCF. 2. The HCF of two numbers divides their LCM.

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

    B. 2 only

    LCM is at least as large as HCF.

  35. Consider the statements about co-prime numbers. 1. Co-prime numbers must both be prime. 2. Two co-prime numbers have HCF 1.

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

    B. 2 only

    8 and 9 are co-prime but not prime.

  36. Consider the statements about divisibility. 1. Divisibility by 8 depends on the last two digits. 2. A number divisible by 2 and 3 is divisible by 6.

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

    B. 2 only

    Divisibility by 8 depends on the last three digits.

  37. Consider the statements about numbers. 1. Zero is a whole number. 2. Zero is a natural number.

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

    A. 1 only

    Natural numbers start from 1.

  38. Consider the statements about Roman numerals. 1. IX stands for 9. 2. XL stands for 40.

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

    C. Both 1 and 2

    IX = 10 - 1 and XL = 50 - 10.

  39. Consider the statements about co-prime numbers. 1. The HCF of two co-prime numbers is 1. 2. The LCM of two co-prime numbers is their product.

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

    C. Both 1 and 2

    For co-primes, HCF = 1 and so LCM = product.

  40. Consider the statements about numbers. 1. Every odd number is prime. 2. The sum of two primes is always even.

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

    D. Neither 1 nor 2

    9 is odd and not prime; 2 + 3 = 5 is odd.

  41. Match the item with its example. 1. Perfect number 2. Twin primes 3. Co-prime pair

    1. 1 - 17 and 19; 2 - 6; 3 - 14 and 15
    2. 1 - 12; 2 - 14 and 15; 3 - 17 and 19
    3. 1 - 14 and 15; 2 - 12; 3 - 6
    4. 1 - 6; 2 - 17 and 19; 3 - 14 and 15
    Answer

    D. 1 - 6; 2 - 17 and 19; 3 - 14 and 15

    6 is perfect, 17 and 19 are twin primes, and 14 and 15 have HCF 1.

  42. Match each problem with the tool to solve it. 1. Largest equal groups from 24 boys and 36 girls 2. Time when two lights blink together again 3. Smallest number divisible by 4 and 6

    1. 1 - HCF; 2 - HCF; 3 - LCM
    2. 1 - LCM; 2 - HCF; 3 - HCF
    3. 1 - LCM; 2 - LCM; 3 - HCF
    4. 1 - HCF; 2 - LCM; 3 - LCM
    Answer

    D. 1 - HCF; 2 - LCM; 3 - LCM

    Largest equal groups need HCF; repeating events and least common multiples need LCM.

  43. The HCF of the fractions 2/3 and 4/9 is

    1. 2/3
    2. 1/9
    3. 4/3
    4. 2/9
    Answer

    D. 2/9

    HCF of fractions = HCF of numerators / LCM of denominators = 2 / 9.

  44. The LCM of the fractions 2/3 and 4/9 is

    1. 2/9
    2. 8/27
    3. 4/3
    4. 2/3
    Answer

    C. 4/3

    LCM of fractions = LCM of numerators / HCF of denominators = 4 / 3.

  45. A teacher arranges 7 counters in rectangles and finds only one arrangement (1 x 7). This activity shows that 7 is

    1. Prime
    2. Even
    3. Composite
    4. A perfect square
    Answer

    A. Prime

    A single array means exactly two factors, so the number is prime.

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