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SSC-Standard Mathematics for Forest Posts · Chapter 1

Number System, Number Theory, HCF and LCM, Logarithms

What to remember

  • HCF × LCM = product of the two numbers. HCF divides all numbers; LCM is divisible by all of them. HCF of fractions = HCF of numerators / LCM of denominators; LCM of fractions = LCM of numerators / HCF of denominators.
  • Divisibility rules and unit digits solve many questions without long division. The count of divisors comes from prime factorisation: if N = aᵖ × bᵠ × cʳ, the number of divisors is (p+1)(q+1)(r+1).
  • Logarithm laws: log(mn) = log m + log n; log(m/n) = log m − log n; log mⁿ = n log m. Remember log₁₀2 = 0.3010 and log₁₀3 = 0.4771.

1. Types of numbers

TypeMeaningExample
NaturalCounting numbers1, 2, 3, ...
WholeNatural numbers and 00, 1, 2, ...
IntegersWhole numbers and negatives−2, 0, 5
Rationalp/q with q ≠ 03/4, 0.25, 0.333...
IrrationalCannot be written as p/q√2, π
PrimeExactly two factors, 1 and itself2, 3, 5, 7
CompositeMore than two factors4, 6, 9
Co-primeHCF of the pair is 1(8, 15)

Key points: 2 is the only even prime. 1 is neither prime nor composite. Zero is an even integer. There are 25 primes below 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. To test whether a number n is prime, divide by all primes up to √n.

Decimals as fractions. A terminating decimal: 0.375 = 375/1000 = 3/8. A pure recurring decimal: 0.ab with bar = ab/99 (for example 0.ababab... = ab/99, so 0.3636... = 36/99 = 4/11). A mixed recurring decimal: 0.1666... = (16 − 1)/90 = 15/90 = 1/6.

2. Divisibility rules

DivisorRule
2Last digit even
3Sum of digits divisible by 3
4Last two digits divisible by 4
5Last digit 0 or 5
6Divisible by both 2 and 3
8Last three digits divisible by 8
9Sum of digits divisible by 9
11(Sum of digits at odd places) − (sum at even places) is 0 or a multiple of 11
7, 11, 13Subtract the number formed by the last three digits from the rest, and test the difference

Example: 5,184 → digit sum 18 → divisible by 9 (and 3). 3,718: odd places 8 + 7 = 15, even places 1 + 3 = 4; difference 11, so divisible by 11.

Remainders and unit digits. The unit digit of a power repeats in a cycle: 2 → 2, 4, 8, 6 (cycle of 4); 3 → 3, 9, 7, 1; 7 → 7, 9, 3, 1; 4 → 4, 6; 9 → 9, 1; numbers ending in 0, 1, 5, 6 keep their unit digit. Example: unit digit of 7²³ → 23 mod 4 = 3, so third in the cycle = 3.

Dividend = (Divisor × Quotient) + Remainder.

3. Factors, squares and counting

  • Prime factorisation: 360 = 2³ × 3² × 5.
  • Number of factors = (3+1)(2+1)(1+1) = 24.
  • Sum of factors = (2⁴ − 1)/(2 − 1) × (3³ − 1)/(3 − 1) × (5² − 1)/(5 − 1) = 15 × 13 × 6 = 1,170.
  • A number is a perfect square if every prime has an even power; then the number of factors is odd.
  • Number of trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ... Example: 25! has 5 + 1 = 6 zeros.
  • Sum of first n natural numbers = n(n+1)/2; sum of first n odd numbers = n²; sum of first n even numbers = n(n+1); sum of squares = n(n+1)(2n+1)/6.

4. HCF and LCM

HCF (GCD) is the largest number that divides all the given numbers. LCM is the smallest number divisible by all of them.

Methods:

  • Prime factorisation: HCF takes the lowest powers of common primes; LCM takes the highest powers of all primes.
  • Division method (Euclid): repeated division until remainder is 0; the last divisor is the HCF.
  • Relationship: for two numbers a and b, HCF × LCM = a × b. (This does not hold for three numbers.)
  • HCF of two co-prime numbers is 1 and their LCM is their product.

Worked examples:

  • 1. HCF and LCM of 12 and 18: 12 = 2² × 3, 18 = 2 × 3². HCF = 2 × 3 = 6; LCM = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18.
  • 2. If HCF = 4 and LCM = 48 and one number is 12, the other = 4 × 48 / 12 = 16.
  • 3. Greatest number dividing a, b, c leaving remainders: when 60, 75 and 90 are divided, the same remainder r: largest divisor = HCF of the differences.
  • 4. Largest number dividing 245 and 1,029 leaving remainder 5 in each case: HCF of (245 − 5) and (1,029 − 5) = HCF (240, 1,024) = 16.
  • 5. Smallest number divisible by 6, 8 and 12: LCM = 24.
  • 6. Smallest number that leaves remainder 3 when divided by 6, 8 and 12: LCM + 3 = 27.
  • 7. Smallest number which when increased by 6 is divisible by 12, 18 and 36: LCM = 36, number = 36 − 6 = 30.
  • 8. Bells ringing together: Bells ring every 12, 15 and 20 minutes together at 8:00; they ring together again after LCM = 60 minutes.
  • 9. Largest square tiles for a floor 6 m × 4 m: side = HCF (600, 400) cm = 200 cm; number of tiles = (600 × 400)/(200 × 200) = 6.
  • 10. Fractions: HCF of 2/3, 4/9 and 6/27 = HCF (2, 4, 6) / LCM (3, 9, 27) = 2/27; LCM = LCM (2, 4, 6) / HCF (3, 9, 27) = 12/3 = 4.
  • 11. Ratio problem: numbers are in ratio 3 : 4 and their HCF is 5, so the numbers are 15 and 20 and the LCM is 60.

5. Surds and roots (gap-fill)

  • √2 = 1.414, √3 = 1.732, √5 = 2.236.
  • Squares 11² to 30² and cubes to 15³ should be known: 13² = 169, 17² = 289, 19² = 361, 12³ = 1,728.
  • Square root by factors: √1,764 = 42 because 1,764 = 2² × 3² × 7².
  • Rationalising: 1/(√a − √b) = (√a + √b)/(a − b).

6. Logarithms

Definition. If aˣ = N (a > 0, a ≠ 1), then x = log_a N. So log₂8 = 3 because 2³ = 8.

Laws

LawFormula
Productlog(mn) = log m + log n
Quotientlog(m/n) = log m − log n
Powerlog mⁿ = n log m
Base changelog_a b = log b / log a
Identitieslog_a 1 = 0; log_a a = 1; a^(log_a x) = x
Reciprocallog_a b × log_b a = 1

Common (base 10) logarithms. log 2 = 0.3010, log 3 = 0.4771, log 5 = 1 − log 2 = 0.6990, log 10 = 1.

Worked examples:

  • 1. log 6 = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781.
  • 2. log 12 = 2 log 2 + log 3 = 0.6020 + 0.4771 = 1.0791.
  • 3. log 1.5 = log 3 − log 2 = 0.1761.
  • 4. log₃ 81 = 4 since 3⁴ = 81.
  • 5. log₂ (1/8) = −3.
  • 6. If log x = 2 then x = 100; if log₂ x = 5 then x = 32.
  • 7. Number of digits in 2¹⁰: 10 × 0.3010 = 3.010, so the characteristic is 3 and the number has 4 digits (1,024).
  • 8. Value of log 8 / log 2 = 3 log 2 / log 2 = 3.

Characteristic and mantissa. For a number greater than 1, characteristic = (number of digits before decimal) − 1. The mantissa is the decimal part and is always positive.

7. Quick problem strategies

  • Check divisibility first; use prime factorisation for factors, HCF and LCM.
  • For "remainder same" problems, use HCF of differences.
  • For "leaves remainder r in each case", use LCM + r; for "short by k", use LCM − k.
  • In log questions, express everything in the prime bases 2, 3 and 5.

Exam traps

  • HCF × LCM = product holds for exactly two numbers.
  • The HCF of fractions has LCM of denominators (not HCF).
  • Number of factors uses (power + 1) for each prime.
  • 1 is neither prime nor composite; 2 is the only even prime.
  • log (m + n) is not log m + log n.
  • log (m/n) is not (log m)/(log n).
  • A negative characteristic is used only for numbers below 1.
  • "Leaves remainder r" uses LCM + r; "when increased by r is divisible" uses LCM − r.

One-liners

  • 1. HCF × LCM = product of two numbers.
  • 2. Divisibility by 11: difference of odd-place and even-place digit sums is 0 or a multiple of 11.
  • 3. 2 is the only even prime.
  • 4. 360 has 24 factors.
  • 5. Trailing zeros of 25! = 6.
  • 6. log 5 = 1 − log 2 = 0.6990.
  • 7. log_a 1 = 0 and log_a a = 1.
  • 8. log 12 = 1.0791.
  • 9. 0.333... = 1/3.
  • 10. 0.1666... = 1/6.
  • 11. The unit digit of powers of 2 cycles through 2, 4, 8, 6.
  • 12. Sum of first n odd numbers = n².

Practice questions

  1. Which of the following is the only even prime number?

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

    C. 2

    2 has exactly two factors, 1 and 2, and is even.

  2. The number 1 is:

    1. Neither prime nor composite
    2. Prime
    3. Composite
    4. Both prime and composite
    Answer

    A. Neither prime nor composite

    A prime needs exactly two factors; 1 has only one.

  3. Which number is divisible by 9?

    1. 5,183
    2. 5,185
    3. 5,187
    4. 5,184
    Answer

    D. 5,184

    Digit sum of 5,184 is 18, divisible by 9.

  4. Which number is divisible by 11?

    1. 3,712
    2. 3,718
    3. 3,716
    4. 3,719
    Answer

    B. 3,718

    Odd-place sum 3+1 = 4, even-place sum 7+8 = 15; difference 11.

  5. The unit digit of 7^23 is:

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

    B. 3

    Cycle of 7 is 7, 9, 3, 1; 23 mod 4 = 3, so the third is 3.

  6. The unit digit of 2^50 is:

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

    D. 4

    Cycle 2,4,8,6; 50 mod 4 = 2, so 4.

  7. The number of factors of 180 is:

    1. 16
    2. 24
    3. 18
    4. 12
    Answer

    C. 18

    180 = 2^2 x 3^2 x 5, so (3)(3)(2) = 18.

  8. The number of factors of 144 is:

    1. 15
    2. 16
    3. 12
    4. 18
    Answer

    A. 15

    144 = 2^4 x 3^2, so (5)(3) = 15.

  9. The sum of all the factors of 12 is:

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

    B. 28

    1+2+3+4+6+12 = 28.

  10. The number of trailing zeros in 50! is:

    1. 12
    2. 13
    3. 11
    4. 10
    Answer

    A. 12

    50/5 = 10 and 50/25 = 2, total 12.

  11. The HCF of 84 and 126 is:

    1. 84
    2. 21
    3. 14
    4. 42
    Answer

    D. 42

    84 = 2^2x3x7 and 126 = 2x3^2x7; HCF = 2x3x7 = 42.

  12. The LCM of 84 and 126 is:

    1. 504
    2. 42
    3. 252
    4. 168
    Answer

    C. 252

    LCM = 84 x 126 / 42 = 252.

  13. The LCM of 12, 16 and 24 is:

    1. 96
    2. 72
    3. 48
    4. 24
    Answer

    C. 48

    12 = 2^2x3, 16 = 2^4, 24 = 2^3x3; LCM = 2^4x3 = 48.

  14. The HCF of 36, 48 and 60 is:

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

    D. 12

    Common lowest powers: 2^2 x 3 = 12.

  15. The product of two numbers is 1,620 and their HCF is 9. Their LCM is:

    1. 14,580
    2. 180
    3. 90
    4. 270
    Answer

    B. 180

    LCM = 1,620 / 9 = 180.

  16. The HCF of two numbers is 4, their LCM is 48 and one number is 12. The other number is:

    1. 16
    2. 8
    3. 24
    4. 12
    Answer

    A. 16

    Other = 4 x 48 / 12 = 16.

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

    1. 24
    2. 51
    3. 27
    4. 30
    Answer

    C. 27

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

  18. The smallest number which when increased by 6 is exactly divisible by 12, 18 and 36 is:

    1. 30
    2. 6
    3. 36
    4. 42
    Answer

    A. 30

    LCM = 36; 36 - 6 = 30.

  19. The greatest number that divides 245 and 1,029 leaving remainders 5 and 4 respectively is:

    1. 15
    2. 3
    3. 10
    4. 5
    Answer

    D. 5

    HCF of 240 and 1,025 is 5.

  20. The greatest number that divides 1,657 and 2,037 leaving remainders 6 and 5 is:

    1. 131
    2. 127
    3. 129
    4. 124
    Answer

    B. 127

    HCF of 1,651 and 2,032 is 127.

  21. Three bells ring at intervals of 8, 12 and 15 minutes. After how many minutes will they ring together again?

    1. 180 minutes
    2. 60 minutes
    3. 120 minutes
    4. 90 minutes
    Answer

    C. 120 minutes

    LCM of 8, 12 and 15 is 120.

  22. The largest square tile that exactly fits a floor 6 m by 4 m has side:

    1. 200 cm
    2. 400 cm
    3. 300 cm
    4. 100 cm
    Answer

    A. 200 cm

    HCF of 600 and 400 is 200.

  23. Two numbers are in the ratio 3 : 4 and their HCF is 5. Their LCM is:

    1. 20
    2. 15
    3. 120
    4. 60
    Answer

    D. 60

    Numbers are 15 and 20; LCM = 60.

  24. The HCF of 2/3, 4/9 and 6/27 is:

    1. 6/27
    2. 2/27
    3. 12/3
    4. 2/9
    Answer

    B. 2/27

    HCF of numerators 2, 4, 6 is 2; LCM of denominators 3, 9, 27 is 27.

  25. The LCM of 2/3, 4/9 and 6/27 is:

    1. 2/27
    2. 12
    3. 4
    4. 36/27
    Answer

    C. 4

    LCM of numerators = 12; HCF of denominators = 3; 12/3 = 4.

  26. The value of 0.3636... (recurring) as a fraction is:

    1. 4/11
    2. 3/8
    3. 9/25
    4. 36/100
    Answer

    A. 4/11

    0.3636... = 36/99 = 4/11.

  27. The value of 0.1666... as a fraction is:

    1. 1/5
    2. 16/99
    3. 1/7
    4. 1/6
    Answer

    D. 1/6

    (16 - 1)/90 = 15/90 = 1/6.

  28. The value of 1 + 2 + 3 + ... + 50 is:

    1. 1,250
    2. 1,275
    3. 1,300
    4. 2,550
    Answer

    B. 1,275

    n(n+1)/2 = 50 x 51 / 2 = 1,275.

  29. The sum of the first 20 odd natural numbers is:

    1. 400
    2. 210
    3. 380
    4. 420
    Answer

    A. 400

    Sum of first n odd numbers is n^2 = 400.

  30. Which of the following is an irrational number?

    1. 22/7
    2. Square root of 49
    3. 0.25
    4. Square root of 2
    Answer

    D. Square root of 2

    Root 2 cannot be written as p/q.

  31. The value of log 2 + log 3 + log 5 (base 10), using log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, is:

    1. 1.3010
    2. 1.4771
    3. 1.0
    4. 0.7781
    Answer

    B. 1.4771

    The sum equals log 30 = 0.3010 + 0.4771 + 0.6990 = 1.4771.

  32. The value of log2 (1/8) is:

    1. 3
    2. -8
    3. -3
    4. 1/3
    Answer

    C. -3

    2^-3 = 1/8.

  33. If log (base 3) x = 4, then x is:

    1. 12
    2. 27
    3. 64
    4. 81
    Answer

    D. 81

    x = 3^4 = 81.

  34. The value of log 8 / log 2 is:

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

    A. 3

    log 8 = 3 log 2, so the ratio is 3.

  35. The value of log 12 (base 10), given log 2 = 0.3010 and log 3 = 0.4771, is:

    1. 0.7781
    2. 1.0791
    3. 1.3010
    4. 0.9542
    Answer

    B. 1.0791

    log 12 = 2(0.3010) + 0.4771 = 1.0791.

  36. log 1.5 (base 10), given log 2 = 0.3010 and log 3 = 0.4771, is:

    1. 0.4771
    2. 0.7781
    3. 0.1761
    4. 0.3010
    Answer

    C. 0.1761

    log 3 - log 2 = 0.1761.

  37. The number of digits in 2^10 using log 2 = 0.3010 is:

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

    B. 4

    10 x 0.3010 = 3.010, characteristic 3, so 4 digits (1,024).

  38. The value of log (base 5) 125 is:

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

    D. 3

    5^3 = 125.

  39. The value of log (base a) 1 is:

    1. 0
    2. 1
    3. a
    4. Undefined
    Answer

    A. 0

    a^0 = 1.

  40. The relation log (base a) b x log (base b) a equals:

    1. 0
    2. a x b
    3. 1
    4. a + b
    Answer

    C. 1

    By change of base, the product is 1.

  41. Consider the statements: 1. HCF x LCM = product of two numbers. 2. This relation holds for any three numbers also. Which is/are correct?

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

    A. 1 only

    The relation is true for two numbers only.

  42. Consider the statements: 1. log (m + n) = log m + log n. 2. log (mn) = log m + log n. Which is/are correct?

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

    B. 2 only

    Logarithms of a product add; log of a sum does not split.

  43. Consider the statements: 1. 2 is the only even prime. 2. 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

    A. 1 only

    1 is neither prime nor composite.

  44. Consider the statements: 1. A perfect square has an odd number of factors. 2. Every prime power in its factorisation is even. 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 true of perfect squares.

  45. How many prime numbers are there below 100?

    1. 24
    2. 26
    3. 20
    4. 25
    Answer

    D. 25

    The primes below 100 number 25, from 2 to 97.

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