HCF (Highest Common Factor) and LCM (Least Common Multiple) concepts are the foundation of many mathematical operations and are essential in solving complex problems. HCF and LCM problems challenge your ability to find the greatest common factor and the smallest common multiple of numbers, and they require both logical and mathematical skills. So get ready to exercise your brain as we delve into the world of HCF and LCM problems and explore the exciting ways they can be used to solve challenging aptitude questions!
What is HCF (Highest Common Factor)?
The Highest Common Factor (HCF) of two numbers is the highest possible number that divides both numbers completely. The Highest Common Factor (HCF) is also known as the Greatest Common Divisor (GCD) or the Greatest Common Factor (GCF).
How to Find HCF?
There are three methods to calculate the HCF of two numbers:
- HCF by Listing Factors Method
- HCF by Prime Factorization
- HCF by Division Method
HCF by Listing Factors Method
In this method, we list the factors of each number and find the common factors of those numbers. Then, among the common factors, we determine the highest common factor.
Example: Find the HCF of 32 and 14.
Answer: First, list down the factors of 32 and 14.
- The factors of 32 are: 1, 2, 4, 8, 16, 32
- The factors of 14 are: 1, 2, 7, 14
We can see that 1 and 2 are the only common factors of 32 and 14, with 2 being the greatest among them. Hence, the HCF of 32 and 14 is 2.
HCF by Prime Factorization
This method involves finding the prime factors of the given numbers.
Example: Find the HCF of 80 and 90.
Answer:
- The prime factors of 80 are: 2 × 2 × 2 × 2 × 5
- The prime factors of 90 are: 2 × 3 × 3 × 5
The common prime factors are 2 and 5. The HCF of 80 and 90 is the product of these common prime factors, which is 2 × 5 = 10.
HCF by Division Method
The HCF of two numbers can be calculated using the division method.
Example: Find the HCF of 30 and 42.
Answer: To find the HCF of 30 and 42, we use the division algorithm:
- Divide 42 by 30, remainder is 12.
- Divide 30 by 12, remainder is 6.
- Divide 12 by 6, remainder is 0.
The last non-zero remainder is 6, hence the HCF of 30 and 42 is 6.
What is LCM (Least Common Multiple)?
In arithmetic, the LCM or least common multiple of two numbers a and b, is denoted as LCM(a, b) and is the smallest or least positive integer that is divisible by both a and b.
How to Find LCM?
There are three methods to find the least common multiple of two numbers:
- LCM by Listing Method
- LCM by Prime Factorization Method
- LCM using Division Method
LCM by Listing Method
We list the multiples of two or more numbers and find their common multiples. The smallest common multiple is the LCM.
Example: Find the LCM of 3 and 4.
Answer:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24…
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28…
The smallest common multiple is 12. Hence, LCM(3, 4) = 12.
LCM by Prime Factorization Method
This method involves finding the prime factors of the given numbers and using them to calculate the LCM.
Example: Find the LCM of 120 and 300.
Answer:
- Prime factors of 120: 2 × 2 × 2 × 3 × 5 = 2^3 × 3^1 × 5^1
- Prime factors of 300: 2 × 2 × 3 × 5 × 5 = 2^2 × 3^1 × 5^2
The highest power of each prime factor is: 2^3, 3^1, 5^2. Hence, the LCM = 2^3 × 3^1 × 5^2 = 8 × 3 × 25 = 600.
LCM by Division Method
In this method, we divide the numbers by common prime factors and use these prime factors to calculate the LCM.
Example: Find the LCM of 3 and 4.
Answer:
- Divide 3 and 4 by the smallest prime number, 2 (not a factor of 3).
- Continue dividing by primes until all numbers become 1.
The LCM is the product of all prime numbers used in division. Hence, LCM(3, 4) = 12.
Finding HCF and LCM of Fractions
For fractions:
- HCF of fractions: HCF of Numerators / LCM of Denominators.
- LCM of fractions: LCM of Numerators / HCF of Denominators.
Example: Find the HCF and LCM of 1/3, 8/7, and 9/11.
Answer:
- LCM of Numerators (1, 8, 9): LCM(1, 8, 9) = 72
- HCF of Denominators (3, 7, 11): HCF(3, 7, 11) = 1
- LCM of Fractions: 72/1 = 72
- HCF of Fractions: HCF(1, 8, 9) / LCM(3, 7, 11) = 1/231
Questions on HCF
Question 1: Find the greatest number that will divide 72, 96, and 120 leaving the same remainder in each case.
Answer:
- Differences: 96 – 72 = 24, 120 – 96 = 24
- HCF of 24 and 24 is 24.
Therefore, the greatest number that will divide 72, 96, and 120 leaving the same remainder is 24.
Question 2: If the HCF of two numbers is 12 and their LCM is 360, find the numbers.
Answer:
- HCF × LCM = a × b
- 12 × 360 = a × b
- 4320 = a × b
Numbers could be 144 and 30 since 144 × 30 = 4320 and HCF(144, 30) = 12.
Question 3: Find the HCF of 36, 48, and 72.
Answer:
- Prime factorization of 36: 2^2 × 3^2
- Prime factorization of 48: 2^4 × 3
- Prime factorization of 72: 2^3 × 3^2
Common factors: 2^2 × 3 = 12.
Question 4: What is the largest three-digit number that is exactly divisible by the HCF of 24 and 36?
Answer:
- HCF of 24 and 36 is 12.
- Largest three-digit number divisible by 12 is 996.
Question 5: The sum of two numbers is 1001, and their HCF is 7. Find the numbers.
Answer:
- Sum: a + b = 1001
- HCF(a, b) = 7
Possible numbers are 504 and 497.
Questions on LCM
Question 1: Find the LCM of 12, 18, and 24.
Answer:
- Prime factorization:
- 12 = 2^2 × 3
- 18 = 2 × 3^2
- 24 = 2^3 × 3
LCM = 2^3 × 3^2 = 72.
Question 2: The LCM of two numbers is 360, and their HCF is 24. If one of the numbers is 120, find the other number.
Answer:
- HCF × LCM = a × b
- 24 × 360 = 120 × b
- b = 72.
Question 3: A factory manufactures products in batches of 16, 24, and 32 units. What is the minimum number of units the factory needs to produce so that each batch can be formed exactly?
Answer:
- LCM of 16, 24, and 32 = 2^5 × 3 = 96.
Solved Problems on HCF and LCM
Problem 1: Two numbers are in the ratio of 5:11. If their HCF is 7, find the numbers.
Solution:
- Numbers: 5m and 11m, HCF = 7.
- Numbers are 35 and 77.
Problem 2: Find the length of the plank which can measure exactly the lengths 4 m 50 cm, 9 m 90 cm, and 16 m 20 cm.
Solution:
- Convert lengths to cm: 450 cm, 990 cm, 1620 cm.
- HCF(450, 990, 1620) = 90 cm.
Problem 3: Find the greatest number which on dividing 70 and 50 leaves remainders 1 and 4 respectively.
Solution:
- HCF of 69 and 46 is 23.
Word Problems on HCF and LCM
Problem 1: The policemen at three different places blow a whistle after every 42 sec, 60 sec, and 78 sec respectively. If they all blow the whistle simultaneously at 9:30:00 hours, when will they whistle again together?
Solution:
- LCM(42, 60, 78) = 5460 sec.
- They will whistle together again at 11:01:00 hours.
Problem 2: A rectangular field of dimension 180m x 105m is to be paved by identical square tiles. Find the size of each tile and the number of tiles required.
Solution:
- HCF(180, 105) = 15m.
- Number of tiles = (180 × 105) / (15 × 15) = 84 tiles.
FAQs
What is the relationship between LCM and HCF?
There’s an inverse relationship between LCM and HCF. LCM(a,b)×HCF(a,b)=a×b\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times bLCM(a,b)×HCF(a,b)=a×b for any two positive integers a and b.
How do you find the HCF of two or more numbers?
The HCF (Highest Common Factor), also known as the GCD (Greatest Common Divisor), of two or more numbers is the largest positive integer that divides all the given numbers without leaving a remainder.
Can the LCM of two numbers be smaller than either of the numbers?
No, the LCM of two numbers cannot be smaller than either of the numbers. It’s always equal to or greater than the largest number among them.
Can the HCF of two numbers be greater than either of the numbers?
No, the HCF of two numbers cannot be greater than their smallest number.
What is the LCM and HCF of coprime (relatively prime) numbers?
When two numbers are coprime, their HCF is 1, and their LCM is the product of the two numbers.
Can LCM and HCF be negative numbers?
No, LCM and HCF are always positive integers, even if the given numbers are negative.
By mastering the concepts of HCF and LCM and practicing with various questions, you can enhance your problem-solving skills and ace your aptitude tests.
