HCF and LCM -theory; meanings, methods and formulae

First we need to understand the meanings of factors and multiples.
Factors and multiples;


When a number a divides another number b exactly, we say that a is a factor of b. In this case, b is known as the multiple of a.

Highest Common Factor(HCF) or Greatest Common Measure(GCM) or Greatest Common Divider(GCD);
The HCF of two or more than two numbers is the greatest number that divides each of them exactly.
2 methods to find the HCF of a set of numbers:
1.Factorisation method: Express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors given HCF. 
2.Division method: Assume that we have to calculate the HCF of two given numbers. Now, we divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required HCF.

Finding the HCF of more than 2 numbers: Sometimes, we'll have to find the HCF of 3 numbers. Then, HCF of [(HCF of any two) and (the third number)] gives the HCF of three given numbers.
Similarly, the HCF of more than 3 numbers can also be obtained using the above mentioned method.

Least Common Multiple(LCM);
The least number which is exactly divisible by each one of the given numbers is called their LCM.
2 methods;
1.Factorisation method of finding the LCM: Resolve each one of the given numbers into a product of prime factors. Then, LCM, is the product of highest powers of all the factors.
2.Common Division method(Short cut method) of finding LCM: Arrange the numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required LCM of the given numbers.

The product of 2 numbers is the product of their HCF and LCM.

2 numbers are said to be co primes if their HCF is 1.

HCF and LCM of fractions are represented by;
1.HCF = HCF of numerators / LCM of denominators
2.LCM = LCM of numerators / HCF of denominators
Decimal fractions;
In given numbers, make the same number of decimal places by annexing(add) zeros in some numbers, if necessary. Considering these numbers without decimal point, find HCF or LCM as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

Comparison of fractions;
Find the LCM of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with LCM as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.