Number system, HCF and LCM questions are the most common “easy” Maths questions in Lok Sewa Paper I. The methods are short, and once you know a few rules you can solve them in under a minute. This note covers the types of numbers, divisibility rules, HCF and LCM methods and the standard word problems, each with a worked example.
1. Types of numbers
| Type | Meaning | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, 4 … |
| Whole numbers | Natural numbers and 0 | 0, 1, 2, 3 … |
| Integers | Whole numbers and negatives | … −2, −1, 0, 1, 2 … |
| Rational numbers | Can be written as p/q (q ≠ 0) | 3/4, −5, 0.25, 0.333… |
| Irrational numbers | Cannot be written as p/q; decimals never end or repeat | √2, √3, π |
| Real numbers | All rational and irrational numbers | — |
| Prime numbers | Exactly two factors: 1 and itself | 2, 3, 5, 7, 11, 13 … |
| Composite numbers | More than two factors | 4, 6, 8, 9, 10 … |
| Co-prime numbers | HCF is 1 | 8 and 15 |
- 1 is neither prime nor composite.
- 2 is the only even prime number.
- There are 25 prime numbers between 1 and 100.
2. Divisibility rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 3,458 |
| 3 | Sum of digits divisible by 3 | 5,271 (sum 15) |
| 4 | Last two digits divisible by 4 | 7,316 (16) |
| 5 | Last digit 0 or 5 | 8,345 |
| 6 | Divisible by both 2 and 3 | 5,274 |
| 8 | Last three digits divisible by 8 | 9,120 (120) |
| 9 | Sum of digits divisible by 9 | 4,572 (sum 18) |
| 10 | Last digit 0 | 6,730 |
| 11 | Difference between sums of alternate digits is 0 or a multiple of 11 | 5,918 (5−9+1−8 = −11) |
3. HCF (Highest Common Factor)
HCF is the largest number that divides all the given numbers exactly.
Method 1, prime factors: 36 = 2² × 3², 48 = 2⁴ × 3. Take the lowest power of each common prime: 2² × 3 = 12.
Method 2, division: divide the larger by the smaller and keep dividing by the remainder. 48 ÷ 36 leaves 12; 36 ÷ 12 leaves 0. HCF = 12.
4. LCM (Lowest Common Multiple)
LCM is the smallest number that is exactly divisible by all the given numbers. Take the highest power of every prime: for 36 and 48, LCM = 2⁴ × 3² = 144.
Another example: 12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3, so LCM = 2³ × 3² = 72.
5. Important formulas
| Rule | Formula |
|---|---|
| Two numbers | HCF × LCM = First number × Second number |
| HCF of fractions | HCF of numerators ÷ LCM of denominators |
| LCM of fractions | LCM of numerators ÷ HCF of denominators |
| Numbers in ratio a : b with HCF h | Numbers are ah and bh; LCM = a × b × h (when a and b are co-prime) |
| 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) |
Check: 12 × 144 = 1,728 and 36 × 48 = 1,728. ✔
6. Standard word problems
| Question type | Use | Example |
|---|---|---|
| Bells, lights or runners meeting together again | LCM | Bells ring every 6, 8 and 12 minutes. LCM = 24, so they ring together after 24 minutes. |
| Largest number that divides a, b and leaves remainder r | HCF of (a − r), (b − r) | 43 and 91 with remainder 7: HCF(36, 84) = 12. |
| Smallest number that, divided by x, y, z, leaves remainder r | LCM + r | 12, 15, 20 with remainder 5: LCM 60 + 5 = 65. |
| Smallest n-digit number divisible by given numbers | Next multiple of the LCM | Smallest 4-digit number divisible by 12, 15, 18: LCM 180, so 180 × 6 = 1,080. |
| Largest tile, rope or container size | HCF | Ropes of 48 m and 36 m cut into equal largest pieces: 12 m. |
📌 Key points to remember
- 1 is neither prime nor composite; 2 is the only even prime; 25 primes up to 100.
- HCF: lowest powers of common primes. LCM: highest powers of all primes.
- HCF × LCM = product of the two numbers.
- “Together again” means LCM; “largest equal size” means HCF.
- Sum of 1 to n = n(n+1)/2; sum of first n odd numbers = n².
Test yourself: Number System, HCF, LCM and Divisibility Rules
Practise Types of numbers, LCM (Lowest Common Multiple) and Standard word problems with 50 questions. Each attempt picks 25 at random.