**1. Numbers:- ****As we all know numbers ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.**

**As we all know numbers ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.**

**Natural Number:** **Counting numbers are called natural numbers. (****Thus, 1, 2, 3, ……..are all-natural numbers.)**

**Counting numbers are called natural numbers. (**

**Thus, 1, 2, 3, ……..are all-natural numbers.)**

**2.****Whole Number:**** All counting numbers, together with 0, form the set of whole numbers. (****Thus, 0, 1, 2, 3, 4, ……… are all whole numbers. )**

**All counting numbers, together with 0, form the set of whole numbers. (**

**Thus, 0, 1, 2, 3, 4, ……… are all whole numbers. )**

**3. Integers: ****All counting numbers, zero, and negative of counting numbers from the set of integers. ( ****Thus, ……….., -3, -2, -1, 0, 1, 2, 3, …….are all integers. )**

**All counting numbers, zero, and negative of counting numbers from the set of integers. (**

**Thus, ……….., -3, -2, -1, 0, 1, 2, 3, …….are all integers. )**

**4. Even numbers:**** A counting number divisible by 2.****( Thus, 0, 2, 4, 6, 8, 10, …………. etc. are all even numbers. )**

**A counting number divisible by 2.**

**( Thus, 0, 2, 4, 6, 8, 10, …………. etc. are all even numbers. )**

**5. ****Odd numbers:** *A counting number is not divisible by 2 is called an odd number*. **( Thus, 1, 3, 5, 7, ………etc. are all odd numbers.)**

**5.**

*A counting number is not divisible by 2 is called an odd number*.

**( Thus, 1, 3, 5, 7, ………etc. are all odd numbers.)**

**6. Prime numbers:**** A counting number is prime if has exactly two factors, itself and 1.**

**A counting number is prime if has exactly two factors, itself and 1.**

* ***( Like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59…….etc )**

**( Like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59…….etc )**

**7. Composite Numbers: ****All counting number which is not prime, are called a composite number.**

**7. Composite Numbers:**

**All counting number which is not prime, are called a composite number.**

*A composite has more than two factors.*

**8. Co-prime: ****Two numbers whose HCF is 1 are called co-prime numbers.**

*Ex. (2, 3), (8, 9) are pairs of co-prime.*

*9. ***Rational Numbers: Numbers that can be expressed in the form of p/q, whose p and q are integers and q≠0, are called rational numbers.**

*Ex. 1/8, 2/5, 0, 6, 9/8 etc.*

**10. Irrational Numbers:-**** Numbers which when expressed in decimal would be in the non-terminating and non-repeating forms are called irrational numbers. **

**Ex. √2, √3, √5, √7, Π, **

**POSITIVE AND NEGATIVE NUMBERS **

*A positive number is a real number that is greater than zero. A negative number is a real number that is smaller than zero. *

*Zero is not positive, nor negative.*

**Multiplication:**

**positive * positive = positive****positive * negative = negative****negative * negative = positive**

**Division: **

**positive / positive = positive****positive / negative = negative****negative / negative = positive**

**Prime Numbers**

A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise, a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1. A number** n>1** is prime if it cannot be written as a product of two factors** a** and **b**, both of which are greater than 1: n = ab.

• The first twenty-six prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

• Note: only positive numbers can be primes.

• There are infinitely many prime numbers.

• The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.

• All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2, and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form** 6n-1** or **6n+1** , because all other numbers are divisible by 2 or 3.

• Any nonzero natural number **n** can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

• Prime factorization: every positive integer greater than 1 can be written as a product of one or more prime integers in a way that is unique. For instance integer **n **with three unique prime factors a, b and c can be expressed as** n = a ^{p}*b^{q}*c^{r}**, where

**p, q,**and

**r**are powers of

**a, b, and c**, respectively, and are >1.

* Example: 4200 = 2^{3}*3*5^{2}*7 *.

• **Verifying the primality** (checking whether the number is a prime) of a given number **n** can be done by trial division, that is to say, dividing by all integer numbers smaller than **√n**, thereby checking whether **n** is a multiple of ** m<√n**.

Example: *Verifying the primality of 161:√161 is little less than 13, from integers from 2 to 13, 161 is divisible by 7, hence 161 is not prime. *

• If **n** is a positive integer greater than 1, then there is always a prime number **P** with **n<p<2n**.

**Factor**

A divisor of an integer ** n**, also called a factor of

*n*, is an integer that evenly divides

*without leaving a remainder. In general, it is said*

**n***is a factor of*

**m***n*, for non-zero integers

**m and n,**if there exists an integer

*such that*

**k****.**

*n = km*• 1 (and -1) are divisors of every integer.

• Every integer is a divisor of itself.

• Every integer is a divisor of 0, except, by convention, 0 itself.

• Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

• A positive divisor of * n* which is different from

*is called a proper divisor.*

**n**• An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.

• Any positive divisor of n is a product of prime divisors of n raised to some power.

• If a number equals the sum of its proper divisors, it is said to be a perfect number. Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:

• If a is a factor of b and a is a factor of c, then a is a factor of **(b+c)**

. In fact, *a* is a factor of **(mb+nc)** for all integers **m and n.**

• If **a** is a factor of **b** and** b** is a factor of **c**, then **a** is a factor of **c**.

• If **a** is a factor of **b** and **b **is a factor of **a**, then **a=****b** or **b=a**.

• If **a** is a factor of **bc**, and **gcd(a,b)=1**, then a is a factor of **c**.

• If **p **is a prime number and **p **is a factor of **ab **then **p** is a factor of **a** or **p **is a factor of **b**.

**Finding the Number of Factors of an Integer **

First make prime factorization of an integer **n=a ^{p}*b^{q}*c^{r}**, where

**a**,

**b**, and

**c**are prime factors of

**n**and

**P**,

**q**, and

**r**are their powers.

The number of factors of **n **will be expressed by the formula **(p+1)(q+1)(r+1)**. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: **450=2 ^{1}*3^{2}*5^{2}**

Total number of factors of 450 including 1 and 450 itself is factors.**(1+1)*(2+1)*(2+1)= 18 factors**

**Finding the Sum of the Factors of an Integer**

First make prime factorization of an integer n = a^{p}*b^{q*}c^{r}^{,} where a, b, and c are prime factors of and p, q, and r are their powers.

The sum of factors of **n** will be expressed by the formula:

Example: Finding the sum of all factors of 450:**450=2 ^{1}*3^{2}*5^{5}**

The sum of all factors of 450 is

**Greatest Common Factor (Divisor) – GCF (GCD)**

The greatest common divisor (GCD), also known as the greatest common factor (GCF), or highest common factor (HCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).

• Every common divisor of a and b is a divisor of GCD (a, b)

• a*b=GCD(a, b)*lcm(a, b)

**Perfect Square**

A perfect square is an integer that can be written as the square of some other integer. For example, 16=4^2 is a perfect square.

There are some tips about the perfect square:

** • The number of distinct factors of a perfect square is ALWAYS ODD.**

** • The sum of distinct factors of a perfect square is ALWAYS ODD. **

**• A perfect square ALWAYS has an ODD number of Odd-factors and EVEN number of Even-factors. **

**• Perfect square always has an even number of powers of prime factors.**

**Divisibility Rules**

**2 – If the last digit is even, the number is divisible by 2. **

**3 – If the sum of the digits is divisible by 3, the number is also. **

**4 – If the last two digits form a number divisible by 4, the number is also. **

**5 – If the last digit is a 5 or a 0, the number is divisible by 5. **

**6 – If the number is divisible by both 3 and 2, it is also divisible by 6. **

**7 – Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7. **

**8 – If the last three digits of a number are divisible by 8, then so is the whole number. **

**9 – If the sum of the digits is divisible by 9, so is the number. **

**10 – If the number ends in 0, it is divisible by 10. **

**11 – If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.**

** Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11. **

**12 – If the number is divisible by both 3 and 4, it is also divisible by 12. **

**25 – Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25**

**Factorials**

Factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For instance 5! = 1*2*3*4*5

• Note: 0!=1.

• Note: The factorial of negative numbers is undefined.

Trailing zeros: **Trailing zeros are a sequence of 0’s in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. **

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer **n**, can be determined with this formula:

n/5+n/5^{2}+n/5^{3}………+n/5^{k},, where k must be chosen such that 5^{k}<1.

**Finding the number of powers of a prime number P, in the n!. **

The formula is: *n/p+n/p ^{2}+n/p^{3}……..till p^{x}<n.*

What is the power of 2 in 25!?

25/2+25/4+25/16 = 12+6+3+1 = 22