Prime Factorization
A prime factor is a factor of a number that is a prime number (a number that has only two factors: 1 and itself).
To find the prime factorization of a number, we break it down into prime numbers only.
Example 1: Prime Factorization of 30
- Step 1: Start dividing by the smallest prime number (2, 3, 5, 7, …).
- Step 2: 30 ÷ 2 = 15 (since 2 is a prime number)
- Step 3: 15 ÷ 3 = 5 (since 3 is also prime)
- Step 4: 5 is already a prime number.
So, prime factorization of 30 = 2 × 3 × 5.
LCM (Least Common Multiple)
LCM of two or more numbers is the smallest number that is a multiple of both.
Example 2: LCM of 4 and 6
- Multiples of 4: 4, 8, 12, 16, 20, 24, …
- Multiples of 6: 6, 12, 18, 24, 30, …
- The smallest common multiple = 12
So, LCM(4,6) = 12
Shortcut using Prime Factorization:
Find the prime factors of both numbers:
- 4 = 2 × 2
- 6 = 2 × 3
- LCM = Take the highest powers of all prime numbers
LCM = 2² × 3 = 12
HCF (Highest Common Factor)
HCF of two numbers is the largest number that can divide both without leaving a remainder.
Example 3: HCF of 18 and 24
- Factors of 18: 1, 2, 3, 6, 9, 18
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Common factors: 1, 2, 3, 6
- The largest common factor = 6
So, HCF(18,24) = 6
Shortcut using Prime Factorization:
- 18 = 2 × 3 × 3
- 24 = 2 × 2 × 2 × 3
- HCF = Take the lowest powers of common prime factors
HCF = 2 × 3 = 6
Relation between LCM and HCF
For two numbers A and B:
LCM(A, B)×HCF(A, B)=A×B
Quick Summary
| Concept | Definition | Example |
|---|---|---|
| Prime Factorization | Breaking a number into prime numbers | 30 = 2 × 3 × 5 |
| LCM (Least Common Multiple) | Smallest number that is a multiple of both | LCM(4,6) = 12 |
| HCF (Highest Common Factor) | Largest number that divides both exactly | HCF(18,24) = 6 |
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