How to Find Lcm Using Prime Factorization Steps and Solved Examples
A whole number higher than 1 whose only factors are 1 and itself is referred to as a prime number. A whole number that may be split evenly into another number is referred to as a factor. 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29 are the first few prime numbers. Any number can be represented as a sum of prime numbers using prime factorisation.
LCM is short for the Least Common Multiples of two or more numbers. As the name suggests, it's a function that finds the least common multiple between two numbers. So, how do you figure out what LCM is? Find out how to find LCM by Prime Factorization Method and in between the article LCM by prime factorisation method worksheet has been added to check the understanding in the article below.
Methods of Prime Factorization
Using the Tree Method
Use any number from the provided number's factor pair to make two branches, and if a factor is prime, circle it.
If a factor is not prime, continue by writing it as the product of a factor pair.
The sum of circled primes is a composite number.
Factor Tree of 36
Using the Ladder Method
The given number is divided by the smallest prime possible and is divided until the number stops dividing.
Keep going until the quotient is prime.
The sum of all the primes at the bottom, sides, and top of the ladder is the composite number.
LCM (Least Common Factor)
A number that is a multiple of both numbers is referred to as a common multiple of two numbers.
Full Form of LCM
Let's say we're looking for multiples of 10 and 25 that are frequently seen. The first few multiples of each integer can be listed. The next step is to find multiples that appear on both lists; these are known as common multiples.
10: 10, 20, 30, 40, 50, 60, 70, 80,90, 100, 110, 120
25: 25, 50, 75, 100, 125
50 and 100 can be seen in both rankings. They are regular multiples of the numbers 10 and 25. If we kept on with the list of multiples for each, we would discover more frequent multiples.
The least common multiple (LCM) is the smallest number which is a multiple of two numbers (LCM). Thus, 50 is the lowest LCM for 10 and 25.
Steps to Calculate LCM by Listing Method
List the initial range of each number's multiples.
Search for multiples that appear on both lists. Write out additional multiples for each number if there aren't any common multiples in the lists.
Find the least quantity on both lists, i.e., LCM number.
How to Find LCM by Prime Factorization Method
Applying the prime factors of two numbers is another method for determining the least common multiple. The steps to be followed to calculate the LCM by the prime factorisation method are as follows:
Finding each number's prime factorisation is the first step in computing the LCM using the prime factors approach.
When you write each number as a product of primes, try to align the primes vertically.
Bring each column's primes down.
To obtain the LCM, multiply the variables.
Below are a few LCM examples with answers and practice problems given on the LCM by the prime factorisation method.
LCM Examples with Answers
Example 1. Calculate the LCM of 15, 20 and 60
Ans: Steps to be followed to calculate LCM are :
1. Finding each number's prime factorisation-
Therefore, 15 = 3, 5
20 = 2, 2, 5
60 = 2, 2, 3, 5
Try to align the primes vertically.
15 = 3, 5
20 = 2, 2, 5
60 = 2, 2, 3, 5
2. Bring each column's primes down.
Therefore- 2, 2, 3, and 5 will be the numbers
3. To obtain the LCM, multiply the variables
Multiplying $2\times 2\times 3\times 5 = 60$
Therefore, LCM will be 60.
Example 2. Calculate the LCM of 50 and 100.
Ans: Steps to be followed to calculate LCM are-
Finding each number's prime factorisation-
$50 = 2\times 5\times 5$
$100 = 2\times 2\times 5\times 5$
Try to align the primes vertically
$50 = 2\times 5\times 5$
$100 = 2\times 2\times 5\times 5$
Bring each column's primes down.
Therefore-2,5, 5 will be the numbers
To obtain the LCM, multiply the variables
Multiplying $2\times 5\times 5=50$
LCM by Prime Factorization Method Worksheet:
Q 1. Calculate the LCM of 15 and 35.
Ans: 105
Q 2. What will be the LCM of 55, 88?
Ans: 440
Q 3. Find the LCM of 60, and 72.
Ans: 360
Summary
In the given article, we have discussed calculating LCM by the prime factorisation method. Therefore, first, we learned about the meaning of prime factorisation and methods of factorisation. Later LCM was calculated by using the prime factorisation method. An LCM by prime factorisation method worksheet is also given to test your knowledge.
FAQs on Lcm By Prime Factorization Method Explained Clearly
1. What is LCM by prime factorization method?
The LCM by prime factorization method is a way of finding the Least Common Multiple by expressing each number as a product of prime numbers and multiplying the highest powers of all primes involved.
- Step 1: Write each number as a product of prime factors.
- Step 2: Identify the highest power of each prime.
- Step 3: Multiply those highest powers.
2. How do you find the LCM using prime factorization step by step?
To find the LCM using prime factorization, factor each number into primes and multiply the greatest powers of each prime.
- Example: Find LCM of 12 and 18.
- 12 = 2² × 3
- 18 = 2 × 3²
- Take highest powers: 2² and 3²
- LCM = 2² × 3² = 4 × 9 = 36
3. What is the formula for LCM using prime factorization?
The formula for LCM using prime factorization is: LCM = product of the highest powers of all prime factors present in the numbers. If a = p₁^a × p₂^b and b = p₁^c × p₂^d, then LCM = p₁^max(a,c) × p₂^max(b,d). This method ensures every prime factor is included with its greatest exponent.
4. Why does the prime factorization method work for finding LCM?
The prime factorization method works because the LCM must contain each prime factor in its highest required power. A number is a common multiple only if it includes all prime factors of the given numbers. Taking the maximum exponent of each prime guarantees the smallest number divisible by all the given numbers.
5. Can you give an example of finding LCM of three numbers by prime factorization?
Yes, the LCM of three numbers is found by taking the highest powers of all prime factors in all three numbers.
- Find LCM of 8, 12, and 15.
- 8 = 2³
- 12 = 2² × 3
- 15 = 3 × 5
- Highest powers: 2³, 3¹, 5¹
- LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120
6. What is the difference between LCM and HCF in prime factorization?
The difference is that LCM takes the highest powers of prime factors, while HCF takes the lowest powers.
- LCM = product of greatest exponents of common and uncommon primes.
- HCF (Highest Common Factor) = product of smallest exponents of only common primes.
7. What are common mistakes when finding LCM by prime factorization?
Common mistakes include missing prime factors or not choosing the highest power of each prime.
- Not fully breaking numbers into prime factors.
- Ignoring a prime that appears in only one number.
- Taking smaller exponents instead of the greatest exponent.
8. Is prime factorization the best method to find LCM?
The prime factorization method is one of the most reliable methods for finding LCM, especially for small and medium numbers. It is particularly useful when numbers are not very large and when learning number theory concepts like prime factors, multiples, and divisibility. For very large numbers, division methods may be quicker.
9. How do you find the LCM of co-prime numbers using prime factorization?
The LCM of co-prime numbers is simply the product of the numbers because they share no common prime factors. For example, 8 = 2³ and 15 = 3 × 5 have no common primes. Therefore, LCM = 2³ × 3 × 5 = 120, which equals 8 × 15. This happens because co-prime numbers have HCF = 1.
10. Where is LCM by prime factorization used in real life?
LCM by prime factorization is used to find the smallest common time, quantity, or cycle when events repeat.
- Finding when traffic lights will turn green together.
- Scheduling repeating events.
- Solving fraction addition with different denominators.
