The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers have a direct relationship: the product of the HCF and LCM of two numbers is equal to the product of the two numbers themselves.
Understanding the Relationship
Let's break down this relationship and illustrate it with an example.
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HCF (Highest Common Factor): Also known as the Greatest Common Divisor (GCD), it is the largest number that divides both given numbers without leaving a remainder.
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LCM (Least Common Multiple): It is the smallest number that is a multiple of both given numbers.
The Key Relationship Explained
The fundamental relationship is:
HCF(a, b) LCM(a, b) = a b
Where 'a' and 'b' are the two numbers.
Example
Let's consider the numbers 12 and 18.
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Finding the HCF: The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The highest common factor is 6. So, HCF(12, 18) = 6.
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Finding the LCM: The multiples of 12 are 12, 24, 36, 48, 60,... The multiples of 18 are 18, 36, 54, 72,... The least common multiple is 36. So, LCM(12, 18) = 36.
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Verifying the relationship:
- HCF(12, 18) LCM(12, 18) = 6 36 = 216
- 12 * 18 = 216
Therefore, HCF(12, 18) LCM(12, 18) = 12 18.
Why This Relationship Matters
This relationship is useful because:
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It provides a way to calculate the LCM if you know the HCF, or vice versa. If you have calculated either the HCF or the LCM, you can use the equation to find the other.
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It helps in simplifying fractions and solving number theory problems. Understanding this connection can be advantageous when dealing with fractions or tackling more complex mathematical problems.
Important Note
This relationship holds only for two numbers. It does not generally apply to three or more numbers. For example, if you have three numbers, a, b, and c, it's not true that HCF(a, b, c) LCM(a, b, c) = a b * c.
