If a number $'a'$ divides another number $'b'$  exactly, then we say that $'a'$ is a factor of $'b'$  and $'b'$  is a multiple of $'a'$.

Factor: A factor of a number is an exact divisor of that number.

Multiple: A number is said to be a multiple of any of its factors.

Example: When $20$ is divided by $5$, the remainder is zero, i.e. $5$ divides $20$ exactly. So $5$ is a factor of $20$ and $20$ is a multiple of $5$.

Even Numbers: All the multiples of $2$ are called even numbers.

Example: $2, 4, 6, 8, 10, 12 \ldots$  are all even numbers.

Odd Numbers: Numbers which are not multiple of $2$ are called odd numbers.

Example: $1, 3, 5, 7, 9, 11, 13, 15 \ldots$  are all odd numbers.

Prime Numbers: A number which has exactly two factors, namely, $1$ and the number itself, is called a prime number.

Example: $2, 3, 5, 7, 11, 13, 17, 19 \ldots$   are all prime numbers.

Composite Number: A number which has more than two factors is called a composite number.

Example: $4, 6, 8, 9, 10, 12, 15 \ldots$   are all composite numbers.

Note:
(a) $1$ is neither prime nor composite. It is the only factor that has one factor, namely itself.
(b) $2$ is the smallest prime number.
(c) $2$ is the only even prime number.

Twin Primes: Two consecutive odd prime numbers are known as twin primes.

Example of Twin Primes: (i)  $3, 5$     (ii)  $5, 7$     (iii)  $11, 13$

Prime Triplets: A set of three consecutive prime numbers, differing by $2$, is called a prime triplet.

Example:  The only prime triplet is $\{3, 5, 7 \}$

Perfect Numbers: If the sum of all the factors of a number is twice the number, then the number is called a perfect number.

Example:

• $6$ is perfect number. Its factors are $1, 2, 3$ and $6$. We see that $1 + 2 + 3 + 6 = 12 = (2 \times 6)$
• Another example is $28$. Its factors are $1, 2, 4, 7, 14$ and $28$. We see that $1 + 2 + 4 + 7 + 14 + 28 = 56 = (2 \times 28)$

Co-Primes:  Two numbers are said to be co-primes if they have no common factor other than 1.

Examples: (i) $3, 4$     (ii) $4, 9$

Note:
a) Two prime numbers are always co-prime
b) Two co-primes are not necessarily prime numbers.

Test of Divisibility

Test of Divisibility by $3$: A number is divisible by $3$ if the sum of the digits is divisible by $3$.

Examples:

• Take a number $814728$. Add the digits $8 + 1 + 4 + 7 + 2 + 8 = 30$. $30$ is divisible by $3$. Hence $814728$ is divisible by $3$.
• Take another number $349709$. Add the digits $3 + 4 + 9 + 7 + 0 + 9 = 32$. $32$ is not completely divisible by $3$. Hence $349709$ is not divisible by $3$.

Test of Divisibility by $4$: A number is divisible by $4$ is the number formed by the last two digits is divisible by $4$.

Examples:

• Take a number $23424224$. The last two digits form the number $24$. Hence the number is divisible by $4$.
• Take a number $234242241$. The last two digits form the number $41$. Hence the number is not divisible by $4$.

Test of Divisibility by $5$: A number is divisible by $5$ if the unit digit is either $0$ or $5$.

Examples:

• Take a number $234242245$. The last digit is $5$. Hence the number is divisible by $5$.
• Take a number $2342422451$. The last digit is $1$. Hence the number is not divisible by $5$.

Test of Divisibility by $6$: A number is divisible by $6$ if it is divisible both by $2$ and $3$.

Example:

• Take a number $753222$. The last digit is $2$, hence divisible by $2$. The sum of the digits is $21$. $21$ is divisible by $3$. Hence the number is divisible both by $2$ and $3$. Hence the number is divisible by $6$.

Test of Divisibility by $8$: A number is divisible by $8$, if the number formed by the last $3$ digits is divisible by $8$.

Examples:

• Take a number $293512$. The number formed by the last three digits is $512$. $512$ is divisible by $8$. Hence the number $293512$ is divisible by $8$.
• Take a number $293513$. The number formed by the last three digits is $513$. $513$ is not divisible by $8$. Hence the number $293512$ is divisible by $8$.

Test of Divisibility by $9$: A number is divisible by $9$ if the sum of the digits is divisible by $9$.

Examples:

• Take a number $874728$. Add the digits $8 + 7 + 4 + 7 + 2 + 8 = 36$. $36$ is divisible by $9$. Hence $874728$ is divisible by $9$.
• Take another number $349709$. Add the digits $3 + 4 + 9 + 7 + 0 + 9 = 32$. $32$ is not completely divisible by $9$. Hence $349709$ is not divisible by $9$.

Test of Divisibility by $10$: A number is divisible by $10$, if its unit digit is $0$.

Examples: $10, 20, 3430, 23249860$ are all divisible by $0$.

Test of Divisibility by $11$: A number is divisible by $11$, if the difference between the sum of its digits at odd places and the sum of the digits at even places is either $0$ or a number divisible by $11$.

Examples:

• Take a number $8192657$. The sum of the digits at odd places $= (7 + 6 + 9 + 8) = 30$. The sum of the digits at even places $= (5 + 2 + 1) = 8$. The difference is $22$ which is divisible by $11$. Hence the number $8192657$ is divisible by $11$.
• Take a number $8192656$. The sum of the digits at odd places $= (7 + 6 + 9 + 6) = 29$.  The sum of the digits at even places $= (5 + 2 + 1) = 8$. The difference is $21$ which is not divisible by $11$. Hence the number $8192657$ is not divisible by $11$.

Prime Factors: A factor of a given number is called a prime factor if this is a prime factor.

Examples: All the factors of $21 are 1, 3, 7$ and $21$. Of these, $3$ and $7$ are prime numbers. Therefore $3$ and $7$ are prime factors of $21$.

Prime Factorization: The method of expressing a natural number as a product of prime numbers is call prime factorization or complete factorization of a given number.

Unique Factorization Property: Any natural number can be expressed as a product of a unique collection of prime numbers except for the order of these prime numbers.

Therefore $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$

Common Factors: A number which divides each one of the given numbers exactly, is called a common factor of each of the given numbers.

Example:
The factors of $36$ are $1, 2, 3, 4, 6, 9, 18$ and $36$.
The factors of $42$ are $1, 2, 3, 6, 7, 14, 21$ and $42$.
Hence the common factors of $36$ and $42$ are $1, 2, 3$ and $6$.

Highest Common Factor (H.C.F) or Greatest Common Divisor (G.C.D): The H.C.F or G.D.C of two or more numbers is the greatest number that divides each of the given numbers exactly.

Example:
Let’s take two numbers $36$ and $54$.
The factors of $36$ are $1, 2, 3, 4, 6, 9, 18$ and $36$.
The factors of $54$ are $1, 2, 3, 6, 9, 18, 27$ and $54$.
Therefore, the HCF or $36$ and $54$ is $18$.

Method of finding the H.C.F of a given numbers

Prime Factorization Method

1. Step 1: Express each one of the given numbers as the product of prime factors
2. Step 2: The product of terms containing least powers of common prime factors gives the H.C.F. of the given numbers.

Example: H.C.F of $324$, $288$ and $360$.
First represent each of the given numbers in to prime factors
$324 = 2^2 \times 3^4$
$288 = 2^5 \times 3^2$
$360 = 2^3 \times 3^2 \times 5$
Therefore H.C.F = Product of terms containing the least powers of common prime factors
$= 2^2 \times 3^2 = 36$

Long Division Method

Step 1: Divide the larger number by smaller number
Step 2: Divide the divisor by smaller one
Step 3: Repeat the process of dividing the preceding divisor by the remainder last obtained, till remainder $0$ is obtained.

Applications of H.C.F

1. Co-Prime Numbers: Two natural numbers are said to be co-prime, if their H.C.F is $1$.
2. To reduce a fraction $\frac{p}{q}$ to the simplest form (lowest form), divide each of the numerator and denominator by the H.C.F of $p$ and $q$.

Least Common Multiple (L.C.M): The L.C.M of two or more numbers is the least natural number which is a multiple of each of the given numbers.

Example: Take two numbers say $12$ and $18$
Multiple of $12 = 12, 24, 36, 48 \ldots$
Multiple of $18 = 18, 36, 54 \ldots$
Hence the L.C.M of $12$ and $18$ is $36$

Methods of finding L.C.M of given numbers

Prime Factorization Method

1. Express each one of the numbers as the product of prime factors
2. The product of all the different prime factors each raised to highest power that appears in the prime factorization of any given numbers, gives the L.C.M of the given numbers.

Example: Find L.C.M of $180, 216$ and 324 \$
Convert each of the numbers into prime factors.
$180 = 2^2 \times 3^2 x 5$
$216 = 2^2 \times 3^3$
$324 = 2^2 \times 3^4$
Therefore L.C.M $= 2^3 \times 3^4 \times 5 = 3240$

Common Division Method

1. Arrange the numbers in any order
2. Divide by a number that divides exactly at least two of the given numbers and carry forward the other numbers
3. Repeat Step 2 till no two numbers are divisible by the same number, other than $1$
4. The product of the divisors and the un-divided numbers is the required L.C.M

Hence the L.C.M $= 2 \times 2 \times 2 \times 3 \times 2 \times 3 = 144$

Relation between H.C.F and L.C.M of two numbers

i. Product of two given numbers = Product of their H.C.F and L.C.M

ii. H.C.F and L.C.M of fractions

a. H.C.F of a given fractions = (H.C.F of numerators / L.C.M of the denominators)

b. L.C.M of a given fractions = (L.C.M of numerators / H.C.F of the denominators)