HCF and LCM Explained
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are fundamental concepts in number theory. They are used to simplify fractions, find common de
TL;DR: HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are fundamental concepts in number theory. They are used to simplify fractions, find comm…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
HCF and LCM: HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are fundamental concepts in number theory. They are used to simplify fractions, find common de
HCF and LCM Explained Simply
HCF (Highest Common Factor) and LCM (Lowest Common Multiple) are fundamental concepts in number theory. They are used to simplify fractions, find common de This is a key Mathematics concept for Class 6-7.
Highest Common Factor (HCF)
The HCF, also called Greatest Common Divisor (GCD), is the largest number that divides two or more numbers evenly. To find HCF, list all factors of each number and identify the largest common factor. For example, factors of 12 are 1, 2, 3, 4, 6, 12 and factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, 6 so HCF(12, 18) equals 6. HCF is useful for simplifying fractions: the fraction 12/18 simplifies to 2/3 by dividing both numerator and denominator by 6.
Finding HCF using Prime Factorization
Prime factorization expresses a number as a product of prime numbers. For 12 equals 2 squared times 3 and 18 equals 2 times 3 squared, the HCF is found by taking the lowest power of each common prime factor: 2^1 times 3^1 equals 6. This method works efficiently for large numbers. For three numbers like 24, 36, and 48: 24 equals 2^3 times 3, 36 equals 2^2 times 3^2, 48 equals 2^4 times 3. The HCF equals 2^2 times 3 equals 12.
Least Common Multiple (LCM)
The LCM is the smallest number that is divisible by two or more given numbers. To find LCM, list multiples of each number and find the smallest common multiple. For example, multiples of 4 are 4, 8, 12, 16, 20, 24... and multiples of 6 are 6, 12, 18, 24... The smallest common multiple is 12, so LCM(4, 6) equals 12. LCM is useful for finding common denominators when adding fractions: to add 1/4 plus 1/6, find LCM(4, 6) equals 12, so 3/12 plus 2/12 equals 5/12.
Finding LCM using Prime Factorization
For prime factorization method, take the highest power of each prime factor that appears. For 12 equals 2^2 times 3 and 18 equals 2 times 3^2, the LCM equals 2^2 times 3^2 equals 36. For 24, 36, 48: 24 equals 2^3 times 3, 36 equals 2^2 times 3^2, 48 equals 2^4 times 3. The LCM equals 2^4 times 3^2 equals 144.
Relationship Between HCF and LCM
For any two numbers a and b, there is an important relationship: HCF(a, b) times LCM(a, b) equals a times b. For example, HCF(12, 18) equals 6 and LCM(12, 18) equals 36, so 6 times 36 equals 216 which equals 12 times 18. This relationship helps verify calculations and provides an alternative method to find one value if the other is known. This relationship does not extend to three or more numbers in the same simple form.
Worked Examples
Example 1. Find HCF(24, 36) using prime factorization.
Solution: 24 equals 2^3 times 3. 36 equals 2^2 times 3^2. HCF takes the lowest power of each common factor: 2^2 times 3 equals 4 times 3 equals 12. So HCF(24, 36) equals 12.
Example 2. Find LCM(10, 15) and verify using the relationship with HCF.
Solution: 10 equals 2 times 5. 15 equals 3 times 5. LCM takes the highest power of each factor: 2 times 3 times 5 equals 30. HCF(10, 15) equals 5. Verify: HCF times LCM equals 5 times 30 equals 150 and 10 times 15 equals 150. Correct!
Example 3. Add the fractions 5/12 plus 7/18 by finding a common denominator using LCM.
Solution: Find LCM(12, 18). 12 equals 2^2 times 3. 18 equals 2 times 3^2. LCM equals 2^2 times 3^2 equals 36. Convert: 5/12 equals 15/36 and 7/18 equals 14/36. Sum equals 15/36 plus 14/36 equals 29/36.
Where You See This in Real Life
- A store receives supplies: candles every 6 days, batteries every 9 days. Using LCM(6, 9) equals 18, both arrive together every 18 days.
- Dividing 24 apples and 36 oranges into identical gift baskets: use HCF(24, 36) equals 12 baskets, each with 2 apples and 3 oranges.
- Cutting a 36 cm by 48 cm piece of cloth into square patches: largest square side is HCF(36, 48) equals 12 cm.
- Two bells ring at intervals: one every 8 seconds, one every 12 seconds. Using LCM(8, 12) equals 24, they ring together every 24 seconds.
Common Mistakes Students Make
- Confusing HCF with LCM - HCF is the largest common factor, LCM is the smallest common multiple
- Incorrectly identifying factors or multiples - double-check your list
- Forgetting to use highest power in LCM or lowest power in HCF - the power matters
- Assuming HCF and LCM are always different - if numbers are coprime (HCF equals 1), then LCM equals their product
Frequently Asked Questions
What are coprime numbers?
Two numbers are coprime if their HCF is 1, meaning they share no common factors other than 1. For example, 8 and 9 are coprime because HCF(8, 9) equals 1. Any two consecutive integers are always coprime.
Is there a formula for HCF and LCM?
For two numbers a and b: HCF(a, b) times LCM(a, b) equals a times b. The Euclidean algorithm efficiently finds HCF using repeated division. For multiple numbers, apply the formula pairwise.
Can HCF be larger than LCM?
No, HCF is always less than or equal to LCM. HCF is a common divisor (at most the smaller number) while LCM is a common multiple (at least the larger number). They are equal only when the numbers are the same.