Factors, Multiples, HCF and LCM
Overview
Factors, multiples, HCF and LCM form the backbone of number theory in primary and upper-primary mathematics. These concepts appear directly in UPTET Paper I and Paper II mathematics sections, and also underpin word problems involving time, work, distribution and measurement. A teacher must not only solve such problems quickly but also explain the underlying logic to young learners.
For UPTET, expect questions on identifying prime and composite numbers, applying divisibility rules, finding HCF and LCM through prime factorisation or division method, and solving application-based problems involving LCM (e.g., bells ringing together) or HCF (e.g., largest tile fitting a floor). Mastery here also strengthens your ability to simplify fractions and work with ratios—topics tested elsewhere in the syllabus.
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Key Concepts
- **Factor**: A number that divides another number exactly (without remainder). Example: 4 is a factor of 12.
- **Multiple**: The product of a number and any whole number. Example: 12, 18, 24 are multiples of 6.
- **Prime number**: A natural number greater than 1 with exactly two factors—1 and itself. Examples: 2, 3, 5, 7, 11, 13.
- **Composite number**: A natural number greater than 1 with more than two factors. Examples: 4, 6, 9, 15.
- **1 is neither prime nor composite**; 2 is the only even prime number.
- **Co-prime (relatively prime) numbers**: Two numbers whose HCF is 1. Example: 8 and 15 are co-prime.
- **HCF (Highest Common Factor)**: The largest number that divides two or more numbers exactly. Also called GCD (Greatest Common Divisor).
- **LCM (Least Common Multiple)**: The smallest number that is a multiple of two or more numbers.
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Formulas / Key Facts
| Concept | Formula / Rule | |---------|----------------| | Relationship between HCF and LCM | HCF × LCM = Product of the two numbers (for two numbers only) | | HCF of co-primes | Always 1 | | LCM of co-primes | Product of the two numbers | | HCF ≤ each number; LCM ≥ each number | Always true | | Prime factorisation for HCF | Take the **lowest** power of all **common** prime factors | | Prime factorisation for LCM | Take the **highest** power of all prime factors appearing in any number |
### Divisibility Rules (must memorise)
| Divisor | Rule | |---------|------| | 2 | Last digit is 0, 2, 4, 6 or 8 | | 3 | Sum of digits divisible by 3 | | 4 | Last two digits form a number divisible by 4 | | 5 | Last digit is 0 or 5 | | 6 | Divisible by both 2 and 3 | | 8 | Last three digits form a number divisible by 8 | | 9 | Sum of digits divisible by 9 | | 11 | Difference of sums of alternate digits is 0 or divisible by 11 |