Understanding HCF and LCM
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are two important concepts in mathematics that deal with the factors and multiples of numbers.
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Highest Common Factor (HCF)
HCF (also known as GCD - Greatest Common Divisor) is the largest number that can exactly divide two or more numbers without leaving a remainder. It’s useful when simplifying fractions or finding common denominators.
How to Find the HCF:
1. List the Factors: Write down all the factors of each number.
Example: For 12 and 18:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
2. Identify the Common Factors: Find the factors that are common to both lists.
Common Factors of 12 and 18: 1, 2, 3, 6
3. Choose the Highest: The highest common factor is the largest of these common factors.
HCF of 12 and 18 is 6.
Applications:
Simplifying Fractions: To simplify the fraction , you divide both the numerator and denominator by their HCF, which is 6:
Solving Problems: HCF is useful when you want to evenly divide something into smaller parts without leftovers.
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Lowest Common Multiple (LCM)
LCM is the smallest number that is a multiple of two or more numbers. It’s helpful when adding or subtracting fractions with different denominators or when finding the common time intervals in scheduling problems.
How to Find the LCM:
1. List the Multiples: Write down the first few multiples of each number.
Example: For 4 and 5:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
2. Identify the Common Multiples: Find the multiples that are common to both lists.
Common Multiples of 4 and 5: 20, 40...
3. Choose the Lowest: The lowest common multiple is the smallest of these common multiples.
LCM of 4 and 5 is 20.
Applications:
Adding/Subtracting Fractions: To add , find the LCM of 4 and 5, which is 20. Convert the fractions:
and , so .
Problem Solving: LCM is useful when determining how often events with different cycles coincide (like bus schedules).
Key Differences:
HCF: Focuses on factors (divisors) of numbers. It finds the largest number that can divide all the given numbers.
LCM: Focuses on multiples. It finds the smallest number that all the given numbers can divide into without leaving a remainder.
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