No, the HCF (Highest Common Factor) of two or more numbers is never greater than their LCM (Lowest Common Multiple).
The HCF is the largest number that divides evenly into all the given numbers. The LCM, on the other hand, is the smallest number that is divisible by all the given numbers. By definition, the smallest number divisible by all given numbers must be larger or equal to the largest number that divides into all the given numbers.
Understanding HCF and LCM
- HCF (Highest Common Factor) / GCD (Greatest Common Divisor): The largest number that divides two or more numbers without leaving a remainder.
- LCM (Lowest Common Multiple): The smallest number that is a multiple of two or more numbers.
Example:
Let's take the numbers 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
The HCF of 12 and 18 is 6.
- Multiples of 12: 12, 24, 36, 48, 60...
- Multiples of 18: 18, 36, 54, 72...
The LCM of 12 and 18 is 36.
In this example, HCF (6) < LCM (36).
Key Relationship
For any two positive integers 'a' and 'b', the product of their HCF and LCM is equal to the product of the numbers themselves:
HCF(a, b) LCM(a, b) = a b
This relationship further demonstrates that the LCM cannot be smaller than the HCF. If the LCM were smaller than the HCF, the left side of the equation would be smaller than the product of 'a' and 'b', which contradicts the fundamental relationship.
Generalization
This principle holds true for more than two numbers as well. The LCM will always be greater than or equal to the HCF.
Conclusion
The HCF of two or more numbers can never be greater than their LCM. The LCM will always be either greater than or equal to the HCF. They are equal only when the given numbers are equal.
