Number System
Natural Numbers: Set of counting numbers is called natural numbers. It is denoted by N. where, N = {1, 2, 3,...... ∞}
Even Numbers: The set of all natural numbers which are divisible by 2 are called even numbers. It is denoted by E. Where, E = {2, 4, 6, 8, 10,...... ∞}
Odd Numbers: The set of all natural numbers which are not divisible by 2 are called odd numbers. In other words, the natural numbers which are not even numbers, are odd numbers. i.e., O = {1, 3, 5, 7,........ ∞}
Whole Numbers: When zero is included in the set of natural numbers, then it forms set of whole numbers. It is denoted by W. Where, W = {0, 1, 2, 3,...... ∞}
Integers: When in the set of whole numbers, natural numbers with negative sign are included, then it becomes set of integers. It is denoted by I or Z. I = [- ∞,…-4, -3, -2, -1, 0, 1, 2, 3, 4,….,∞] Integers can further be classified into negative or positive Integers. Negative Integers are denoted by Z and positive Integers are denoted by Z + . Z – {-∞,....., -3, -2, -1} and Z + = {1, 2, 3,......... ∞} Further, 0 is neither negative nor positive integer.
Prime Numbers: The natural numbers which have no factors other than 1 and itself are called prime numbers. Note that, (i) In other words, they can be divided only by themselves or 1 only. As, 2, 3, 5, 7, 11 etc. (ii) All prime numbers other than 2 are odd numbers but all odd numbers are not prime numbers. (iii) Every prime number other than 2, 3 can be represented in the form of 6𝑘 ± 1. But every number that is this form may or may not be a prime number
Co-Prime Numbers: Two numbers which have no common factor except 1, are called Co- Prime numbers. Such as, 9 and 16, 4 and 17, 80 and 81 etc. It is not necessary that two co-prime numbers are prime always. They may or may not be prime numbers.
Divisible numbers/Composite numbers: The whole numbers which are divisible by numbers other than itself and 1 are called divisible numbers or we can say the numbers which are not prime numbers are composite or divisible numbers. As, 4, 6, 9, 15. Note: 1 is neither Prime number nor composite number. Composite numbers may be even or odd.
Rational Numbers: The numbers which can be expressed in the form of 𝑝 𝑞where p and q are integers q and r co-primes and q ≠ 0 are called rational numbers. It is denoted by Q. These may be positive, or negative. e.g. 4 5, 5 1, − 1 / 2 etc are rational numbers.
Irrational Numbers: The numbers which are not rational numbers, are called irrational numbers. Such as √2 = 1.414213562.........., 𝜋 = 3.141592653
Real Numbers: Set of all rational numbers as well as irrational numbers is called Real numbers. The square of all of them is positive.
Cyclic Numbers: Cyclic numbers are those numbers of n digits which when multiplied by any other number up to n gives same digits in a different order. They are in the same line. As 142857 2 × 142857 = 285714 3 × 142857 = 428571 4 × 142857 = 571428 5 × 142857 = 714285
Perfect Numbers: If the sum of all divisors of a number N (except N) is equal to the number N itself then the number is called perfect number. Such as, 6, 28, 496. 8128 etc. The factor of 6 are 1, 2 and 3 1 + 2 + 3 = 6 1 + 2 + 4 + 7 + 14 = 28 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128
In a perfect number, the sum of inverse of all of its factors including itself is 2 always. e.g. Factors of 28 are 1,2,4,7,14 are 1 1 + 1 / 2 + 1 / 4 + 1 / 7 + 1 / 14 + 1 / 28 = 56 / 28 = 2
Complex Numbers: Z = a + ib is called complex number, where a and b are real numbers, b ≠ 0 and i = √- 1. Such as, √-2. √3 etc. So, a + ib or 4 + 5i are complex numbers. Factors:-
The numbers with two factors are prime numbers
The numbers with three factors are squares of prime numbers
The numbers with four factors are cubes of prime numbers or product of two prime numbers
The numbers with five factors are fourth powers of prime numbers.
Perfect squares have odd number of factors
Number after being expressed as N = am × bn × co × dp × …(prime Factorization) where 𝑎, 𝑏, 𝑐, 𝑑 …are prime factors of 𝑁, and 𝑚, 𝑛, 𝑜, 𝑝… are their powers Then the number of factors will be: (𝑚 + 1)(𝑛 + 1)(𝑜 + 1)(𝑝 + 1)….
Number of even factor s = 𝑚(𝑛 + 1)(𝑜 + 1)(𝑝 + 1)….. (Here, m is the power of even prime factors and 𝑛, 𝑜, 𝑝, …are powers of odd prime factors)
Number of odd factors = (𝑛 + 1)(𝑜 + 1)(𝑝 + 1)…. (Here, 𝑛, 𝑜, 𝑝,…. are powers of odd prime factors of the number)
Sum of all the factors = (a0 + a1 + … + am)(b1 + b2 + … + bn)(c1 + c2 + … + co)(d1 + d2 + … + dp)….. Note:- From here we can use the sum of terms in a Geometric progressions to get the sum of terms inside each bracket.
Sum of all the even factors of a number = (a1 + … + am)(b1 + b2 + … + bn)(c1 + c2 + … + co)(d1 + d2 + … + dp)…..; where a is the even prime factor of the number and 𝑏, 𝑐, 𝑑, …. are odd prime factors of the number and 𝑚, 𝑛, 𝑜, 𝑝, … are their powers respectively
Sum of all odd factors of a number = (b 1 + b2 + … + bn)(c1 + c2 + … + co)(d1 + d2 + … + dp)….. where 𝑏, 𝑐, 𝑑,.. are odd prime factors of the number and 𝑛, 𝑜, 𝑝,.. are their powers respectively.
Number of ways in which the number can be represented as a product of two distinct factors: (i) Perfect squares: - 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 − 1 2; 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 + 1 2, if product of same factors is also included (ii) Non-Perfect squares: - 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 2
Product of all factors: - (Number)Number of factors/2
Co-primes: The number of co-primes to N which are less than N = 𝑁(1 − 1 / 𝑎)(1 − 1 / 𝑏)... a, b, are Prime Bases in the prime factorization of N.
Sum of all the Co-primes of N which are less than N is:- 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜 − 𝑝𝑟𝑖𝑚𝑒𝑠 2 × 𝑁
Highest power of ‘P’ which can exactly divide n! is: [𝑛 𝑃] + [𝑛 𝑃2] + [𝑛 𝑃3] + ... + [𝑛 𝑃𝑥]. this formula goes on till we get 𝑛𝑚 − 𝑛𝐶1(𝑛 − 1)𝑚 + 𝑛𝐶2(𝑛 − 2)𝑚 − ..... − 𝑛𝐶𝑛 − 1(1)𝑚 = 1 Note: [] is greatest integer function.
Number of zeroes in n! is equal to highest power of 5 in n!.
Divisibility Rules: - Number Divisibility Rule (DR) 2 Last digit is a multiple of 2 4 Last two digits is a multiple of 4 8 Last three digits is a multiple of 8 16 Last four digits is a multiple of 16 3 Sum of digits is a multiple of 3 9 Sum of digits is a multiple of 9 5 Unit digit should be 5 or 0 10 Unit digit is zero 6 DR of 3 and 2 12 DR of 4 and 3 (we cannot take 2 and 6, as they are not co-primes) 15 DR of 5 and 3 18 DR of 2 and 9 (we cannot take 6 and 3, as they are not co-primes) 11 Difference between two sum of Alternate digits is a multiple of 11 or zero
Power Cyclicity to Find Unit Digit: - Unit Digit Repeated digits Power cyclicity 0 0 1 1 1 1 2 2, 4, 8, 6 4 3 3, 9, 7, 1 4 4 4, 6 2 5 5 1 6 6 1 7 7, 9, 3, 1 4 8 8, 4, 2, 6 4 9 9, 1 2 Last two-digits: -
Last two digits of (Odd)20m = 01; m is a natural number (Exception 5)
Last two digits of a number with 5 is 75 only when ten’s digit of the base is odd and also the power is odd, else the last two digits will be 25.
Last two digits any number N 2 is equal to the last two digits of (50 - N)2, (50 + N)2 and (100- N)2
Last two digits of 210m = 24; if m is odd
Last two digits of 210m = 76; if m is even Remainders:
𝑅 [(𝑁 − 1)! 𝑁] = 𝑁 − 1, where N is a prime number
𝑅 [(𝑁 − 2)! 𝑁] = 1, where N is a prime number
𝑅 [𝑀𝑁 − 1 𝑁] = 1, Where N is a prime number, M and N are pair of Co-primes.
𝑅 [𝑀𝑁 𝑁] = 𝑀, Where N is a prime number, M and N are pair of Co-primes.
The expression an − bn is always divisible by a - b.
The expression an − bn is divisible by a + b only when n is odd.
The expression an + bn is divisible by a + b if n is odd. If n is even, it is not divisible by a + b.