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Study Guide: HCF & LCM
Source: https://www.fatskills.com/quantitative-aptitude-and-numerical-ability-for-competitive-examinations/chapter/hcf-lcm

HCF & LCM

By Fatskills Exam Guides Team — the exam nerds behind 28,500+ quizzes and 2.1M practice questions across 500+ global exams.

⏱️ ~2 min read
LCM (Least Common Multiple) & HCF (Highest Common Factor)
The concept of LCM would find a wide application in a lot of problems in the subsequent chapters. This is an important concept which can be used to simplify many problems.
LCM or Lowest Common Multiple i.e., is the smallest positive number which is a multiple of each of the given numbers. That means which can be divided by all the given numbers. Though 6, 12, 18, 24 can divided 72, 144, 288, 576... but amongst all these numbers 72 is the smallest i.e.,LCM is 72.
HCF or GCD (Greatest Common Divisor) is the largest factor of the given set of numbers. Similarly 24, 48, 72 can be divided by each of 2, 3, 4, 6, 12… but 12 is the greatest common divisor of all these numbers henceHCF or GCD is 12.
LCM/HCF can be found for integers, fraction polynomials, or decimals.

Example: Find the LCM and HCF of 6 and 8

Multiples of 6 are 6, 12, 18, 24, 30, 42, 48… so on. The factors are 1, 2, 3, 6.
Multiples of 8 are 8, 16, 24, 32, 40, 48…. so on. The factors are 1, 2, 4, 8.
Common multiples are: 24, 48…. etc., Least of these is 24.Thus, 24 is the LCM of 6 and 8.
The common factors are 1, 2. The largest of these is 2.So the HCF is 2.
But method is tedious for determining the LCM/HCF of large numbers or more than 2 numbers.

A better method is the 'Factorisation Method'.
In this method, each of the numbers is factorised as:
6 = 2 8 = 2
Example Find the LCM and HCF of 12 and 18.

(1) Divide the large number by the smaller. (2) In the next step divide the previous divisor by the previous remainder.

Continue with the process till you end up with a remainder 0.

The LAST divisor when the remainder becomes 0 is the HCF of the 2 numbers. Applying it to the given problem we get.

If two numbers a and b are given, if their LCM and HCF are L and H respectively, then L
i.e.,product of LCM and HCF of two numbers =product of numbers themselves.


Example Find the LCM and HCF of and LCM =

HCF =

Example Find the LCM and HCF of and .

Simplify both the fractions to their lowest forms i.e.,
Now the LCM
And the HCF
Note: Do not directly apply the formula if the fractions are not in their simplest form. The applications of LCM and HCF galore. You would come across it in a practically the entire arithmetic.

Some Fundamental Applications
Example A heap of coconuts is divided into groups of 2, 3 and 5 and each time no coconut is left over. Find the least number of coconuts in the heap.

Let X be the number of coconuts desired. X must be a common multiple of 2, 3 and 5, otherwise some coconuts will be left over. Further, the number X must be the least possible number. So X has to be the LCM of 2, 3, 5 which is 30. So the least number of coconuts in the heap is 30.



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