In mathematics, the complete form of the term “Least Common Multiple” is “Least Common Multiple,” whereas the whole structure of “Highest Common Factor” is “the Highest Common Factor.”
The highest common factor (HCF) specifies the most significant factor that is present between two or more numbers. In comparison, the lowest common multiple (LCM) indicates the smallest number that is precisely divisible by two or more numbers.
The most significant common factor (GCF) is another name for the highest common factor (HCF), while the least common divisor is another name for the least common multiple.
The prime factorization technique and the division method are two fundamental approaches that may be used to determine the highest common factor and the lowest common multiple, respectively.
Definition of HCF
The greatest common divisor, often known as the gcd of two or more positive integers, is the most significant positive integer used to divide the numbers without producing a remainder.
This is because the principles of arithmetic require that this be the case. Take, for instance, the numbers 8 and 12. The most significant common factor for the numbers 8 and 12 will be four since this is the most important number that can be divided by both 8 and 12.
Definition of LCM
In mathematics, the symbol for the least common multiple (also known as the LCM) of two integers, such as a and b, is “LCM” (a,b). On the other point, the least common multiple is the slightest positive integer that can be divided by both a and b. It is also the lowest positive integer. Take, as an illustration, the positive numbers 4 and 6, for instance.
The numbers 4, 8, 12, 16, 20, and 24 are all multiples of 4.
The numbers 6, 12, 18, and 24 are all multiples of 6.
The numbers 12, 24, 36, 48, and so on all fall into the common multiples of 4 and 6. The number 12 would be the least frequent multiple in that group.
Difference between HCF and LCM
The highest-order common factor, or HCF, is the factor that contributes the most to each of the supplied numbers. On the other hand, the LCM is the multiple of the provided integers that is the lowest of all the common multiples.
The HCF of the above numbers can’t be more significant than any of them. On the other hand, the LCM of the above numbers can’t be less than any of them.
Finding HCF by Factorization
Using the prime factorization method, the first step in determining the highest common factor of the supplied numbers is to determine the numbers’ prime factors. After we have chosen the elements, the next step is to determine the product of the prime factors shared by all the supplied numbers.
For example
HCF of 50 and 75
Factors of 50 = 2 × 5 × 5
Factors of 75 = 3 × 5 × 5
Five times five is a component in the numbers 50 and 75. Therefore, the HCF of (50, 75) equals 25.
Finding LCM by Factorization
Finding the LCM of a set of integers using prime factorization may be done using one of two different ways.
The following steps require to be considered for computing the least common multiple of any set of integers using the prime factorization method:
Create a list of the prime factors of the provided numbers and note the prime factors shared across the numbers.
The lowest common multiple of the provided figures is equal to the product of the figures’ common prime factors and their uncommon prime factors.
For example
LCM of 160 and 90
160 Prime factors = 2 × 2 × 2 × 2 × 2 × 5
90 Prime factors = 2 × 3 × 3 × 5
Common factors = 2 × 5
UnCommon factors = 2 × 2 × 2 × 2 × 3 × 3
LCM = 1440
Real-life uses of HCF
The HCF, or “Highest Common Factor,” is the largest significant integer that can split the provided numbers into two equal parts without leaving a residual. When we want to divide things into smaller portions, we may use it to organize things into rows or groups, hand out larger quantities of stuff to more people at once, calculate how many people we need to invite, and so on.
Practical estimation
You may use the idea of HCF to figure out the same thing if you are scheduling a party and want to ensure that nothing is wasted or if you need a reasonable estimate. Both items can be achieved by using the concept of HCF.
Equal distribution of things
When trying to distribute something evenly over a group or set, HCF is the necessary approach. Consider the scenario where ten children are given twenty chocolates and forty sweets. In this case, we will give each kid two chocolates and four sweets to maintain equality.
Resource optimization
You may use HCF to estimate the minimal number of square towels that can be cut from a range of fabrics with a specified length and breadth without wasting material.
Arrangement of stuff in columns and rows
You can efficiently store two or more quantities of various dimensions by using the highest common factor (HCF) of the amounts; this may be used to determine and implement the implementation.
Real-life uses of LCM
The LCM, or “Least Mutual Multiple,” is the minimum number perfectly divisible by all the numbers provided. It is used to address issues relating to racetracks, and traffic lights, anticipate when an event will occur again over the same period and other similar challenges. The following are some examples of how the LCM principle manifests itself in our day-to-day lives:
Repetition of events and tasks
The timings of the traffic lights have to be adjusted by the traffic controllers so that not all lights are green at the same time, particularly during the busier times of the day. Because of this potential to cause issues, they should only be set for a few minutes.
A traffic controller determines the time for each signal by taking the timings of neighboring spots within the same region and calculating the least common multiple (LCM) of all traffic pauses.
In this manner, he can speedily control the traffic by raising or lowering each movement’s length or setting alarms at certain intervals during the day.
Solving fractions
In arithmetic, the idea of LCM plays an essential role in solving fractions. To decode offenses with different denominators, we need to reduce them to the same common denominator. To do this, finding a common denominator for them via LCM is essential.
Consider the next time something will occur at the same time as before. It is used to determine something that will happen again at the exact moment. For example, it is advantageous to launch rockets and space shuttles toward planets in the closest proximity to the earth.
Encryption
The LCM of significant prime numbers is also beneficial in RSA encryption methods, which are used to transport information through internet connections in a safe manner.
Final Words
When pursuing higher education in mathematics or other related fields, it is beneficial to have a solid understanding of the concepts of LCM and HCF. When planning, estimating, and dividing things, conceptualizing these procedures may be very advantageous.
Kids improve their ability to think critically and solve problems when they are exposed to ideas in the context of real-world applications. Many platforms make it easier for children to understand various mathematical concepts and how they may be used in the real world. It gives children access to different educational materials and interactive games, which helps youngsters develop a mentality for visualizing learning and problem-solving.