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NUMBER SYSTEMS

Number System

Natural 1,2,3,4,5,6,7,……,n
Integers -n, …., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,….,n
Positive 1,2,3,4,5,6,7,……,n
Rational m/n where m and n are integers and n is not zero.
Real M rational number or the limit of a convergent sequence of rational numbers
Complex m+ni where a and b are real numbers and I is the square root of -1

Prime Number

Starting from the basic knowledge, a prime number is a natural number which has only two distinct divisors: 1 and itself.

The number 1 is not a prime number.

There are 25 prime numbers under 100: 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.

Prime Factorization Theorem: This is the area where prime numbers are used. This theorem states that any integer greater than 1 can be written as a unique product of prime numbers.

Examples:

Thus, prime numbers are the basic building blocks of any positive integer. This factorization will also help in finding GCD and LCM quickly.

Perfect Numbers

A number is a perfect number if the sum of its factors, excluding itself and but including 1, is equal to the number itself.

Example: 6 (1 + 2 + 3 = 6), 28 (1 + 2 + 4 + 7 +14 = 28)

Co-Prime Numbers

Two numbers are co-prime to each other, if they do not have any common factor except 1.

Example: 25 and 9, since they don’t have a common factor other than 1

Points to Remember

  1. The number 1 is neither prime nor composite.
  2. The number 2 is the only even number which is prime.
  3. (xn+ yn) is divisible by (x + y), when n is an odd number.
  4. (xn– yn) is divisible by (x + y), when n is an even number.
  5. (xn– yn) is divisible by (x – y), when n is an odd or an even number.

Important Formulas

  1. (a + b)(a – b) = (a2– b2)
  2. (a + b)2= (a2 + b2 + 2ab)
  3. (a – b)2= (a2 + b2 – 2ab)
  4. (a + b + c)2= a2 + b2 + c2 + 2(ab + bc + ca)
  5. (a3+ b3) = (a + b)(a2 – ab + b2)
  6. (a3– b3) = (a – b)(a2 + ab + b2)
  7. (a3+ b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)
  8. When a + b + c = 0, then a3+ b3 + c3 = 3abc
  9. (a + b)n= an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn

Shortcuts for number divisibility check

  1. A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8.
  2. A number is divisible by 3, if the sum of its digits is divisible by 3.
  3. A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
  4. A number is divisible by 5, if its unit’s digit is either 0 or 5.
  5. A number is divisible by 6, if it is divisible by both 2 and 3.
  6. A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
  7. A number is divisible by 9, if the sum of its digits is divisible by 9.
  8. A number is divisible by 10, if it ends with 0.
  9. A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
  10. A number is divisible by 12, if it is divisible by both 4 and 3.
  11. A number is divisible by 14, if it is divisible by 2 as well as 7.
  12. Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

Shortcuts for ‘recurring decimal to fraction’ conversion

  1. For recurring decimals of format ‘0.abababab…’ (ab repeats), equivalent fraction will be “repeating group (here ab)”/”as many 9’s as the number of digits in repeating group”.
  2. For recurring decimals of format ‘0.abbbbb…’ (b repeats), equivalent fraction will be (entire decimal group – non-repeating decimal group)/(as many 9’s as the number of repeating digits in the decimal part with as many 0’s as the number of non-repeating digits in the decimal part).

 

Some Question frequently asking in the exam

  • Given a number x, you will be asked to find the largest n digit number divisible by x.
  • You will be given with a set of numbers (n1, n2, n3…) and asked to find how many of those numbers are divisible by a specified number x.
  • Given a number series, find the sum of n terms, find nth term etc.
  • Find product of two numbers when their sum/difference and sum of their squares is given.
  • Find the number when divisibility of its digits with certain numbers is given.
  • Find the smallest n digit number divisible by x.
  • Which of the given numbers are prime numbers.
  • Number x when divided by y gives remainder r, what will be the remainder when x2 is divided by y.
  • Given relationship between the digits of number, find the number.
  • Find result of operations (additions, subtractions, multiplications, divisions etc) on given integers. These integers can be large and the question may look difficult and time consuming. But mostly the question will map onto one of the known algebraic equations given in this first tab.

Most Important and Very Interesting number calculation rule: PEMDAS