Yes, the Least Common Multiple (LCM) of two or more numbers is always divisible by their Highest Common Factor (HCF).
This fundamental mathematical principle is crucial when dealing with multiples and factors of numbers. Understanding this relationship simplifies complex calculations and offers insights into number theory.
Understanding the Relationship Between LCM and HCF
The Highest Common Factor (HCF) is the largest number that divides two or more numbers exactly. The Least Common Multiple (LCM), on the other hand, is the smallest number that is a multiple of two or more numbers. According to the provided reference, this relationship exists between them:
- "The HCF of two or more numbers is a factor of their LCM." This statement directly confirms that the LCM is always divisible by the HCF.
- "For any 2 natural numbers, their LCM is always divisible by their HCF." This further clarifies that the divisibility is applicable to any two natural numbers.
Illustrative Examples
To further understand this, let's look at a few examples:
Example 1:
| Numbers | HCF | LCM | Divisible? (LCM/HCF) |
|---|---|---|---|
| 12, 18 | 6 | 36 | 36/6 = 6 (Yes) |
| 15, 20 | 5 | 60 | 60/5 = 12 (Yes) |
Example 2:
- Consider the numbers 8 and 12.
- HCF (8, 12) = 4
- LCM (8, 12) = 24
- 24 is divisible by 4 (24/4 = 6).
Example 3:
- Consider the numbers 10, 15 and 20.
- HCF (10, 15, 20) = 5
- LCM (10, 15, 20) = 60
- 60 is divisible by 5 (60/5=12)
Conclusion
In every scenario, the LCM is divisible by the HCF. This is a consistent rule for any combination of two or more natural numbers. As shown, the reference confirms this to be a true mathematical property and this applies across all sets of natural numbers.
