Friday 22 July 2011

NUMBER SYSTEM

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NUMBER SYSTEM:Number Systems forms the base for quant ability and clearing of concepts is important for CAT and other related exams. Following table gives a brief introduction to system of numbers.

number_systems

Prime Number :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:
{550 = 2 \times 5^2 \times 11}


{1200 = 2^4 \times 3 \times 5^2}

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 RememberThe number 1 is neither prime nor composite.

  • The number 2 is the only even number which is prime.
  • (xn + yn) is divisible by (x + y), when n is an odd number.  
  • (xn – yn) is divisible by (x + y), when n is an even number.  
  • (xn – yn) is divisible by (x – y), when n is an odd or an even number.

Factors of a Number:Representing a number as prime factors helps in analyzing problems.

N = p^a + q^b + r^c Where p, q, r are prime numbers and a, b, c are the number of times each prime number occurs.
Number of Factors = (a + 1)(b + 1)(c + 1)

Number of Ways of Expressing a Given Number as a Product of Two Factors

{{(a+1)(b+1)(c+1)} \over 2}
Sum of Factors = {({a^{p+1} - 1})({b^{q+1} - 1})({c^{r+1} - 1}) \over {(a-1)(b-1)(c-1)}}

Concept of Cyclicity :Concept of cyclicity is used to find unit's digit in case the numbers are occuring in powers.

Cyclicity of 1, 5, 6 - 1

Cyclicity of 4, 9 - 2

Cyclicity of 2, 3, 7, 8 - 4

Maximum Power of p (prime nubmer) in n! (n factorial)

{n \over p} + {n \over p^2} + {n \over p^3} + \dots

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