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# Mathematics-Number Theory

**Numbers ::-**A number is denoted by group of digits, i.e., 0 to 9.

**Numeral :: -**Group of digits called numeral.

**Face Value :: -**The face value of digit in a numeral is its own value. ie., in the numeral 456, the face value of 5 is 5.

**Place Value :: -**Place value of a digit is multiplying the particular digit with place where it stands. That means if the digit stands in 10th position, the place value of that digit is digit * 10.

Eg::) In the numeral 23, the place value of 2 is 2 * 10 = 20

**Natural Numbers :: -**Counting numbers are called natural numbers. eg:: 1, 2, 3, 4, 5, 6, . . . .

**Whole Numbers :: -**All counting numbers and 0 included in the whole numbers. eg:: 0, 1, 2, 3, 4, . . . .

**Integers ::-**All counting numbers, zero, and negative of counting numbers are included in the integer section. eg: ..., -3, -2, -1, 0, 1, 2, 3, . . .

**Even Numbers :: -**A counting number divisible by 2 is called an even number. Eg:: 0, 2, 4, 6, 8, . ....

**Odd Numbers ::-**A counting number not divisible by 2 is called an odd number. Eg:: 1, 3, 5, 7, 9 ......

**Primary Numbers :: -**A counting number is called prime numbers, if it has exactly 2 factors. That are number itself and 1. Eg:: ) Prime Number less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

**Test for Prime Number ::**n

^{2 }>= p. If p is divisible by any of the prime number less than or equal to n, then p is not prime, else prime.

Eg::) check whether the number is prime or not. (1) 143 (2) 173

(12)

^{2}>= 143 , Prime number less than 12 are 2, 3, 5, 7, 11. clearly 143 is divisible by 11. So 143 is not a prime number.

(14)

^{2}>= 173, prime numbers less than 14 are 2, 3, 5, 7, 11, 13. Clearly none of them divides 173. So 173 is prime number.

**Composite Numbers ::-**The natural numbers which are not included in the prime numbers are called composite numbers.

**Two natural numbers x & y are said to be co primes, if their HCF is 1.**

Co Primes ::-

Co Primes ::-

**Divisibility by 2 ::-**A number is divisible by 2 if its last digit is any of 0, 2, 4, 6, 8.

**Divisibility by 3 ::-**A number is divisible by 3, if its sum of digits is divisible by 3.

**Divisibilty by 9 ::-**A number is divisible by 9, if its sum of digits is divisible by 9.

**Divisibility by 4 ::-**A number is divisible by 4, if its sum of last 2 digits is divisible by 4.

**Divisibility by 8 ::-**A number is divisible by 8, if number formed by its last 3 digits is divisible by 8.

**Divisibility by 10 ::-**If last digit of a number is 0, then it can by divisible by 10.

**Divisibility by 5 ::-**If last digit of a number is 0 or 5, then it can be divisible by 5.

**Divisibility by 11 ::-**Difference between sum of digits at odd places and sum of digits at even places is either 0, or a number is divisible by 11.