Least
Common Multiple (LCM)
Least Common Multiple (LCM) of two or more
numbers is the least number which is divisible by each of these numbers without
a remainder. The same can be algebraically defined as “LCM of two or more
expressions is the expression of the lowest dimension which is divisible by
each of them without remainder.” .
Highest Common Factor (HCF)
Highest Common Factor (HCF) is the largest
factor of two or more given numbers. The same can be defined algebraically as “HCF
of two or more algebraical expressions is the expression of highest dimension
which divides each of them without remainder. .
HCF is also called GCD (Greatest Common
Divisor). .
Product of two numbers = LCM x HCF
For finding LCM and HCF of fractions, the
following formulae may be remembered:
HCF of fractions = 
LCM of fractions = 
Finding LCM and HCF of given numbers
LCM and HCF can each be found by either one
of two methods:
— Factorization
— Long Division
We will look at both the methods. .
LCM by factorization
Resolve the numbers into prime factors. Then
multiply the product of all the prime factors of the first number by those
prime factors of the second number which are not common to the prime factors of
the first number. .
This product is then multiplied by those
prime factors of the third number which are not common to the prime factors of
the first two numbers. .
In this manner, all the given numbers have to
be dealt with and the last product will be the required LCM. .
Example:
Find the LCM of 108, 144, 270. .
108 = 18 x 6 = 33 x 22
144 = 12 x 12 = 32 x 24
270 = 27 x 10 = 33 x 5 x 2
LCM = 33 x 5 x 24 = 2160
LCM by Division
Select any one prime factor common to at
least two of the given numbers. Write the given number in a line and divide
them by the above prime number. Write down the quotient for every number under
the number itself. If any of the numbers is not divisible by the prime factor
selected, write the number as it is in the line of quotients. .
Repeat this process for the line of quotients
until you get a line of quotients which are prime to each other (i.e., no two “quotients”
should have a common factor). .
The product of all the divisors and the
number, in the last line will be the required LCM. .
Another method is elimination using prime
numbers. .
Example:
Find the LCM of 72, 42, 90. .
7 72, 42, 90
6 72, 6, 90
3 12, 1, 15
4, 1, 5
Therefore, LCM = 7 x 2 x 3 x 3 x 4 x 5 = 2520
HCF by Factorization
Resolve the given number into prime factors.
The product of the prime factors common to all the numbers will be the required
HCF. .
Example:
Find the HCF of 256, 964, 424. .
256 = 16 x 16 = 24 x 24
= 28
964 = 241 x 4 = 22 x 241
424 = 53 x 8 = 23 x 53
Hence HCF = 4
HCF by Long Division
Take two numbers. Divide the greater by the
smaller; then divide the divisor by the remainder, divide the remainder by the
next remainder and so on until the remainder is zero. The last divisor is the
HCF of the two numbers taken. .
By the same method find the HCF of this HCF
and the third number. This will be the HCF of the three numbers.
Example:
HCF of 1241 and 8979
Note: For any two numbers, product of these
numbers = product of their LCM and HCF
Factorial
Factorial is defined for any positive
integer. It is denoted by ∠ or !. Thus “Factorial
n” is written as n! or /n. n! is defined as the product of all the integers
from 1 to n. .
Thus n! = 1.2.3. ... (n - 1), n. .
0! is defined to be equal to 1. .
Therefore 0! = 1 and 1! = 1. .
Remainder Theorem
When a polynomial function f(x) is divided by
(x - a), the remainder is f(a). .
For example, when x2 - 2x + 5 is
divided by x - 1, the remainder will be f (1), i.e. 12 - 2(1) + 5 =
4 .
We can see that if f(x) is divided by (x + a),
then the remainder will be f (-a). .
For example, when x3 + x2
- 5x - 4 is divided by x +1, then the remainder will be f (-1),
i.e., (-1)3 + (-1)2 - 5(-1)
- 4. i.e., 1 .
if f (a) is zero, it means that the remainder
is zero and hence, we can say that (x - a) is a factor of f(x). .
Some important rules
The student should remember the following
very important rules pertaining to xn – yn and xn
+ yn when n is a positive integer. .
Rules pertaining to xn - yn
(i) It
is always divisible by x - y (i.e., x - y is always a factor of xn –
yn).
(ii) It
is also divisible by x + y when n is even. .
(iii) It
is not divisible by x + y when n is odd. .
Rules pertaining to xn + yn
(i) It
is never divisible by x - y (i.e., x - y is never a factor of xn + yn).
.
(ii) It
is divisible by x + y whenever n is odd. .
(iii) It
is not divisible by x + y when n is even. .
These six rules are very useful for certain
problems that are common in various entrance exams. For example, if we have a
number like 19n + 1, since this is of the form of xn + yn,
we can conclude that it is divisible by 20 (= 19 + 1) whenever n is odd (as per
the rules discussed above). Similarly, if we have a number like 13n
- 1, since this is of the form xn – yn, we can conclude
that it is always divisible by 12 (=13 - 1). We can say it is also divisible by
14 (= 13 + 1) whenever n is even (as per the rules discussed above). .
Sum of natural numbers, their squares and cubes
The sum to ‘n’ terms of the following series
are quite useful and hence should be remembered by students. .
Sum of first n natural numbers = Sn = 
Sum of the Squares of first n natural numbers
= Sn2 = 
Sum of the Cubes of first n natural numbers Sn3 = 
