**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) 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). .

For finding LCM and HCF of fractions, the following formulae may be remembered:

HCF of fractions =

LCM of fractions =

LCM and HCF can each be found by either one of two methods:

— Factorization

— Long Division

We will look at both the methods. .

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. .

Find the LCM of 108, 144, 270. .

108 = 18 x 6 = 3^{3} x 2^{2}

144 = 12 x 12 = 3^{2} x 2^{4}

270 = 27 x 10 = 3^{3} x 5 x 2

LCM = 3^{3} x 5 x 2^{4} = 2160

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. .

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

Resolve the given number into prime factors. The product of the prime factors common to all the numbers will be the required HCF. .

Find the HCF of 256, 964, 424. .

256 = 16 x 16 = 2^{4} x 2^{4}
= 2^{8}

964 = 241 x 4 = 2^{2} x 241

424 = 53 x 8 = 2^{3} x 53

Hence HCF = 4

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.

HCF of 1241 and 8979

**Note: For any two numbers, product of these
numbers = product of their LCM and HCF **

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. .

When a polynomial function f(x) is divided by (x - a), the remainder is f(a). .

For example, when x^{2} - 2x + 5 is
divided by x - 1, the remainder will be f (1), i.e. 1^{2} - 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 x^{3} + x^{2}
- 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). .

The student should remember the following
very important rules pertaining to x^{n} – y^{n} and x^{n}
+ y^{n} when n is a positive integer. .

(i)** ** It
is always divisible by x - y (i.e., x - y is always a factor of x^{n} –
y^{n}).

(ii) It is also divisible by x + y when n is even. .

(iii) It is not divisible by x + y when n is odd. .

(i) It
is never divisible by x - y (i.e., x - y is never a factor of x^{n} + y^{n}).
.

(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 19^{n }+ 1, since this is of the form of x^{n} + y^{n},
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 13^{n}
- 1, since this is of the form x^{n }– y^{n}, 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). .

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
= Sn^{2} =

Sum of the Cubes of first n natural numbers Sn^{3} =