This chapter is the base for every other arithmetic topic: kinds of numbers, divisibility tests, unit digits of large powers, counting factors, the order of operations (BODMAS), fractions and recurring decimals, and HCF and LCM with their word problems.
Kinds of Numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers (N) | Counting numbers starting from 1 | 1, 2, 3, … |
| Whole numbers (W) | Natural numbers and zero | 0, 1, 2, … |
| Integers (Z) | Positive and negative whole numbers and zero | …, −2, −1, 0, 1, 2, … |
| Rational numbers (Q) | Can be written as p/q with q ≠ 0; decimals terminate or repeat | ½, −3, 0.75 |
| Irrational numbers | Cannot be written as p/q; decimals neither end nor repeat | √2, √3, π |
| Prime numbers | Exactly two factors: 1 and the number itself | 2, 3, 5, 7, 11 |
| Composite numbers | More than two factors | 4, 6, 8, 9 |
| Co-prime numbers | Two numbers whose HCF is 1 | (8, 9), (14, 15) |
Facts to Remember
- 1 is neither prime nor composite; 2 is the only even prime.
- There are 25 primes from 1 to 100, and 4 primes from 1 to 10 (2, 3, 5, 7).
- Odd × odd = odd; even × any integer = even; odd + odd = even.
- Sum of the first n natural numbers = n(n + 1)/2; sum of their cubes = [n(n + 1)/2]².
Divisibility Rules
| Divisor | Test | Example |
|---|---|---|
| 2 | Last digit even | 3748 |
| 3 | Sum of digits divisible by 3 | 5316 → 15 |
| 4 | Last two digits divisible by 4 | 7312 → 12 |
| 5 | Last digit 0 or 5 | 4835 |
| 6 | Divisible by both 2 and 3 | 4632 |
| 8 | Last three digits divisible by 8 | 54136 → 136 |
| 9 | Sum of digits divisible by 9 | 8127 → 18 |
| 11 | Difference between sums of alternate digits is 0 or a multiple of 11 | 7469 → 13 − 13 = 0 |
| 12 | Divisible by both 3 and 4 | 1452 |
Unit Digit of a Power
| Last digit of base | Cycle of unit digits | Cycle length |
|---|---|---|
| 0, 1, 5, 6 | Always the same digit | 1 |
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
| 4 | 4 (odd power), 6 (even power) | 2 |
| 9 | 9 (odd power), 1 (even power) | 2 |
Method
Divide the power by the cycle length; the remainder picks the digit (remainder 0 → last digit of the cycle)
Example: 7⁹⁵ → 95 ÷ 4 leaves 3 → third digit of (7, 9, 3, 1) = 3.
Factors and Remainders
Number and sum of factors
If N = pᵃ × qᵇ × rᶜ, number of factors = (a + 1)(b + 1)(c + 1)
Sum of factors = (1 + p + … + pᵃ)(1 + q + … + qᵇ)(1 + r + … + rᶜ). Example: 360 = 2³ × 3² × 5 has 4 × 3 × 2 = 24 factors, and their sum is 15 × 13 × 6 = 1170.
- Trailing zeros of n! = [n/5] + [n/25] + [n/125] + … (count of factor 5). Example: 100! ends in 20 + 4 = 24 zeros.
- aⁿ + bⁿ is divisible by (a + b) when n is odd; aⁿ − bⁿ is divisible by (a − b) for every n.
- Remainder of a power: replace the base by its remainder first. Example: 14 leaves 4 on division by 5, and 4² leaves 1, so 14¹⁰⁰ leaves 1.
BODMAS: Order of Operations
Order
Brackets → Of / Orders (powers, roots) → Division and Multiplication → Addition and Subtraction
Division and multiplication have equal priority and are done from left to right; so are addition and subtraction. Brackets are removed in the order: bar (vinculum), ( ), { }, [ ].
Common Trap
20 ÷ 5 × 2 = 4 × 2 = 8, not 20 ÷ 10 = 2. Work left to right when ÷ and × appear together.
Fractions and Recurring Decimals
- To compare a/b and c/d, compare a × d with b × c (cross-multiplication).
- 0.ab̄ (two digits repeating) = ab/99; for example 0.3737… = 37/99. One repeating digit: 0.ā = a/9.
- Mixed decimal: 0.3̄6 type → (number formed by all digits − non-repeating part) ÷ (as many 9s as repeating digits, followed by as many 0s as non-repeating digits). Example: 0.36̄ = (36 − 3)/90 = 11/30.
HCF and LCM
| HCF (highest common factor) | LCM (least common multiple) | |
|---|---|---|
| Meaning | Largest number dividing all the given numbers | Smallest number divisible by all the given numbers |
| By prime factors | Common primes with the lowest powers | All primes with the highest powers |
| Of fractions | HCF of numerators ÷ LCM of denominators | LCM of numerators ÷ HCF of denominators |
| Typical use | Largest tile, biggest container, equal rows | Bells or lights together again, smallest number divisible by several |
Key relation (two numbers only)
HCF × LCM = product of the two numbers
Example: HCF 23, LCM 1449, one number 161 → other = 23 × 1449 ÷ 161 = 207. This does not hold for three numbers.
| Problem type | Answer |
|---|---|
| Largest number dividing a, b, c leaving the same remainder | HCF of the differences (b − a), (c − b), (c − a) |
| Largest number dividing a, b, c leaving remainders p, q, r | HCF of (a − p), (b − q), (c − r) |
| Smallest number leaving the same remainder r when divided by several numbers | LCM + r |
| Smallest number where (divisor − remainder) is the same constant k for every divisor | LCM − k |
Worked Examples
- Simplify 18 ÷ 3 × (4 + 2) − 5². Bracket: 18 ÷ 3 × 6 − 5²; power: 18 ÷ 3 × 6 − 25; left to right: 6 × 6 − 25 = 36 − 25 = 11.
- Three lights turn green every 36, 48 and 72 seconds and are together at 9:00 AM. LCM = 2⁴ × 3² = 144 s = 2 min 24 s, so next together at 9:02:24 AM.
- A courtyard 48 m × 36 m is paved with the largest square tiles. Side = HCF(48, 36) = 12 m; tiles = 4 × 3 = 12.
- Largest number dividing 445, 572 and 699 leaving remainders 4, 5 and 6: HCF(441, 567, 693) = 3² × 7 = 63.
- Smallest 4-digit number divisible by 12, 15, 18 and 27: LCM = 2² × 3³ × 5 = 540; 1000 is not a multiple, the next multiple is 1080.
- Least number which leaves 3 when divided by 5, 6, 7 and 8 but is divisible by 9: LCM = 840, numbers are 840k + 3; k = 2 gives 1683 = 9 × 187.
- Unit digit of 2¹⁰⁰: 100 ÷ 4 leaves 0, so take the last digit of the cycle (2, 4, 8, 6) = 6.
Common Mistakes
Treating 1 as prime; doing × before ÷ instead of left to right; using HCF × LCM = product for three numbers; swapping the fraction rules for HCF and LCM; forgetting that remainder 0 in a unit-digit question means the last digit of the cycle.
Exam Focus
Expect one or two questions on BODMAS simplification, one on HCF/LCM word problems (bells, tiles, remainders) and one on divisibility or unit digit. Memorise squares up to 30 and cubes up to 15 to save time.