Mathematics places a considerable deal of importance on the concepts of HCF and LCM since they are crucial for resolving a wide range of issues involving time and work, time and distance, pipes and cisterns, etc. Calculations can be completed more quickly and with less effort if the LCM and HCF of two or more numbers are known.
Finding the largest tape to measure land, calculating the largest tile size, and other mathematical issues are among those that HCF is very useful. On the other hand, LCM is beneficial for constructing encrypted messages utilising cryptography and for resolving issues with racetracks, traffic signals, and other computer-related issues.
It is crucial to first comprehend the underlying ideas and definitions of LCM and HCF in order to further comprehend their properties.
Table of Contents:
LCM FULL FORM IN MATHS
What is LCM?
The Lowest Common Multiple (LCM) is the smallest multiple that can be shared by at least two natural numbers. Consider the numbers 2, 3, and 4 for instance. Their LCM would be 12.
HCF FULL FORM IN MATHS
What is HCF?
The highest factor that any two or more given natural numbers have in common is referred to as the Highest Common Factor (HCF). The Greatest Common Divisor (GCD) is another name for this. Taking the numbers 24 and 36 into consideration, their HCF would be
Difference Between LCM and HCF
In number theory, the words LCM (Least Common Multiple) and HCF (Highest Common Factor) are employed. The following are the variations between LCM and HCF:
The lowest multiple that all of the numbers shares is known as the LCM of two or more numbers.
The greatest factor that unites all the numbers is the HCF of two or more numbers.
Calculation: To determine the LCM of two or more numbers, we find each number’s multiples until we locate a shared multiple. The LCM is the smallest of these common multiples.
Finding the factors of each number and then identifying the largest factor that is shared by all the numbers allows us to determine the HCF of two or more integers.
Relationship: Insofar as their products are equal to the sum of the two numbers, the LCM and HCF of the two numbers are connected.
For instance, the product of two numbers a and b is ab. Then, LCM(a,b) * HCF(a,b) = ab can be stated.
Meaning: When we need to determine a common multiple of two or more values, we utilise the LCM. When adding or subtracting fractions, for instance, we must identify a common denominator, which is the LCM of the denominators.
When two or more numbers need to have a common factor, HCF is utilised. For instance, in order to simplify fractions, we must identify the numerator and denominator’s largest common factor.
LCM and HCF Formula
It is possible to use a formula to represent the link between the LCM and HCF of two numbers. Finding the HCF and LCM of two numbers “a” and “b” using the following formula:
an x b = HCF(a,b) x LCM(a,b)
According to this formula, the sum of any two numbers’ HCF and LCM is the same as the sum of the two numbers themselves. to learn more about the connection between LCM and HCF.
We can determine the unidentified value using the HCF and LCM formulas.
HCF (a, b) = b LCM (a, b)
Techniques for LCM and HCF
If 4 numbers have an HCF of 5, then their product will equal their LCM. For instance, since the HCF of the pair, 5 and 9 is 5, their LCM will be 5 x 9 = 45.
The HCF for a pair of non-coprime numbers is always 3. Let’s take the non-coprime numbers 6 and 9, for instance. Since they share factor 3, we can see that their HCF is 3.
How to find LCM and HFC?
There are various methods for finding the Least Common Multiple (LCM) and Highest Common Factor (HCF) of numbers. The two most popular techniques are:
Division method using the prime factorization method
Let’s examine these techniques in more detail.
LCM By Prime Factorization Method:
We must first determine the prime factors of the given numbers in order to calculate LCM and HCF using the prime factorization method. Then, by putting the following procedure into practice, we can calculate the values of HCF and LCM:
HCF Prime Factorization Method:
We must identify the prime factors of the given numbers in order to calculate their HCF using prime factorization. We must determine the product of the common prime factors that each of the provided integers shares after determining the factors. In some circumstances, in order to get the HCF of those integers, we must multiply the prime factors with the least power. Let’s use prime factorization to get the HCF for the numbers 48 and 72.
The prime factors of 48 are either 24 3 or 2 2 2 2.
The prime factors of 72 are either 23 32 or 2 2 2 3 3.
2 2 2 2 or 23 is the common factor of 48 and 72. The HCF of (48, 72) is thus equal to 8.
HCF By Division Method:
Write each integer as the sum of its prime factors in step one.
Step 2: To obtain the LCM, multiply the highest powers of all the prime factors that may be found in any of the supplied numbers.
Let’s use prime factorization to determine the LCM of 347 and 829 as an example.
Step 1: Since 347 and 829 are both prime numbers, additional factoring is not possible.
Step 2: To calculate the LCM, we simply multiply 347 by 829 because there are no additional prime factors to take into account.
LCM is 347 divided by 829, which is 287563.
The LCM of 347 and 829 is thus 287563.
List of LCM and HCM Properties
The characteristics of HCF and LCM can be used to evenly distribute groups of things into their greatest grouping or divide objects into smaller pieces. We discover intriguing correlations between the HCF and LCM features as we learn more about them. The following are some crucial traits of HCF and LCM:
Property 01
The HCF is always equal to or less than any of the given numbers for any given set of numbers. As an illustration, consider pairs 7 and 28, where the HCF is 7, which is smaller than both 7 and 28. The numbers 5 and 15 might be used as another illustration. The HCF in this case is 5, which is one of the integers.
By way of illustration, the HCF of the numbers 24, 36, and 48 is 12, which is lower than all of the given numbers.
Property 02
The HCF of the given numbers will always be 1 if they are co-prime. In such circumstances, the product of the supplied co-prime integers and the LCM of those numbers will equal one.
LCM of coprime numbers, thus, equals the product of the numbers.
Think of two co-prime numbers, such as 14 and 15.
LCM = 210 for 14 and 15.
14 and 15 combined provide 210.
LCM (14, 15) is therefore 14 times 15 = 210.
Property 03
The following formula can be used to get the HCF and LCM of fractions:
LCM of fractions is equal to HCF of Denominators / LCM of Numerators.
HCF of fractions is equal to LCM of denominators / HCF of numerators.
Consider the fractions 3/8 and 5/12 as an illustration.
The numerators are 3 and 5, and the denominators are 8 and 12.
LCM (3, 5) = 15
The formula for the (3, 5) = 1 LCM (8, 12) = 24 HCF (8, 12) = 4
Now, using the aforementioned formula:
HCF (3/8, 5/12) = HCF (3, 5) / LCM (8, 12) = 1/24 LCM (3/8, 5/12) = LCM (3, 5) / HCF (8, 12) = 15 / 4
Property 04
Any two or more numbers have an HCF that is either less than or equal to one of the specified values.
Let’s use the numbers 9 and 15 as an example.
The product of 9 and 15’s HCF is 3, which is less than both 9 and 15.
Using the numbers 5 and 25 as another example, the HCF is 5, which is equal to one of the numbers.
As a result, we may state that the HCF of any pair of numbers is always less than or equal to the smaller of the two.
Property 05
Any two or more numbers that make up the LCM must always be greater than or equal to one of the provided values.
Let’s use the numbers 6 and 10 as an example.
30 is the LCM of 6 and 10, which is higher than both 6 and 10.
Another illustration is the LCM of the integers 3 and 9, which equals one of the numbers.
Thus, the greatest number among any pair of numbers or greater is always bigger than or equal to their LCM.
LCM and HFC Examples
Example 1:
Find the 47, 56, and 63 numbers with the highest common factor.
Solution:
Three numbers—47, 56, and 63—are given.
47 is a prime number, as we all know.
56 = 2 × 2 × 2 × 7
63 = 3 × 3 × 7
According to the formula above, the sole factor that all three numbers share is 7.
Example 2:
The LCM for 32, 36, 42, and 48 is what?
Solution:
32, 36, 42, and 48 LCM
Let’s use prime factorization to determine LCM.
32 can be divided into prime factors by 2 to get 25 by 2.
36 is prime factorised as 2 x 2 x 3 x 3 = 22 x 32.
42’s prime factorization is 2 x 3 x 7.
48 is prime factorised as 2 x 2 x 2 x 2 x 3 = 24 x 3.
Thus,
LCM (32, 36, 42, 48) is 2x2x2x2x2x3x3x5x7 = 6048.
Either the prime factorization approach or the long division method can be used to find the LCM and HCF.
Frequently Asked Questions on HCF and LCM
Either the prime factorization approach or the long division method can be used to find the LCM and HCF.
The HCF of two numbers multiplied by the LCM of two numbers is the definition of a product. This means that the LCM of two numbers is equal to the product of two numbers divided by their HCF, and the HCF of two numbers is the same.
Using the prime factorization technique, we can determine the GCF (Greatest Common Factor) of 32 and 48 as follows:
32’s prime factorization is 2 x 2 x 2 x 2 = 25.
48 can be divided into four prime factors, each of which is equal to two.
The highest common factor among the prime factors of both numbers must be found in order to determine the GCF. Since 24 is the largest power of 2 which is a factor of both 32 and 48, it is the highest common factor in this situation.
The GCF of 32 and 48 is therefore 16 (or 24).
