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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 :: n2 >= 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.
Co Primes ::- Two natural numbers x & y are said to be co primes, if their HCF is 1.
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.
