🗺️ Chapter Roadmap
1 Factors vs. Multiples
Before understanding LCM and HCF, it is very important to know the exact difference between a "Factor" and a "Multiple".
✂️ Factors (The Dividers)
These are the smaller numbers that can perfectly divide a larger number (leaving a remainder of 0).
1, 2, 3, 4, 6, 12
(Because 12 comes in the tables of all these numbers)
🏃 Multiples (The Times Table)
These are the numbers you get when you multiply a number by 1, 2, 3, 4, etc. (Reading its table).
5, 10, 15, 20, 25...
(This list goes on forever up to infinity)
2 HCF (Highest Common Factor)
The Highest Common Factor is the largest number that can perfectly divide two or more given numbers.
💡 Real-life Example (Ribbon Cutting)
You have two ribbons. One is 12 cm long and the other is 18 cm long. You want to cut them into equal pieces. What is the longest piece you can cut without wasting any ribbon?
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
Common Factors: 1, 2, 3, 6
Highest Common Factor (HCF) = 6 cm
3 LCM (Least Common Multiple)
The Least Common Multiple is the smallest number that comes in the multiplication table of two or more given numbers.
🐸 Real-life Example (Jumping Frogs)
Two frogs start jumping from the same starting line. Frog A covers 4 meters in one jump. Frog B covers 6 meters in one jump. At what distance will they both land on the exact same spot for the first time?
- Frog A's jumps: 4, 8, 12, 16, 20, 24...
- Frog B's jumps: 6, 12, 18, 24, 30...
Common Meeting Points: 12m, 24m, 36m...
Least Common Multiple (LCM) = 12 meters
4 The Prime Factorization Method
For larger numbers, writing down all factors or multiples is too slow. Instead, we break numbers down into their Prime Numbers (2, 3, 5, 7, etc.).
📝 Example: Find HCF and LCM of 12 and 18
-
1
Step 1: Break them down into prime numbers.12 = 2 × 2 × 3
18 = 2 × 3 × 3 -
2
Step 2: To find HCF, multiply ONLY the matching (common) numbers.Matches: One 2 and one 3.
HCF = 2 × 3 = 6 -
3
Step 3: To find LCM, take the highest amount of each prime number used.The number 2 appears at most twice (in 12).
The number 3 appears at most twice (in 18).
LCM = 2 × 2 × 3 × 3 = 36
5 The Magic Formula
In almost every competitive exam, there is one question directly based on this golden rule. Memorize it!
"The product of LCM and HCF is always equal to the product of the two numbers."
📝 Formula Example:
The LCM of two numbers is 60 and their HCF is 5. If the first number is 15, find the second number.
60 × 5 = 15 × x
300 = 15x
x = 30015
2nd Number = 20