Number System

Number System Study Materials – Quantitative Aptitude

Face Value and Place Value of a Digit:

Face Value: It is the value of the digit itself eg, in 3452, face value of 4 is ‘four’, face value of 2 is ‘two’.

Place Value: It is the face value of the digit multiplied by the place value at which it is situated eg, in 2586, place value of 5 is 5 × 102 = 500.


Number Categories

Natural Numbers (N): If N is the set of natural numbers, then we write N = {1, 2, 3, 4, 5, 6,…}

The smallest natural number is 1.

Whole Numbers (W): If W is the set of whole numbers, then we write W = {0, 1, 2, 3, 4, 5,…}

The smallest whole number is 0.

Integers (I): If I is the set of integers, then we write I = {– 3, –2, –1, 0, 1, 2, 3, …}

Rational Numbers: Any number which can be expressed in the form of p/q, where p and q are both integers and q # 0 are called rational numbers.


Basic Rules on Natural Numbers:

  1. One digit numbers are from 1 to 9. There are 9 one digit numbers. ie, 9 × 100.
  2. Two digit numbers are from 10 to 99. There, are 90 two digit numbers. ie, 9 × 10.
  3. Three digit numbers are from 100 to 199. There are 900 three digit numbers ie, 9 × 102.

In general the number of n digit numbers are 9 × 10(n–1)

Sum of the first n, natural numbers ie, 1 + 2 + 3 + 4 + … + n = n n 1 / 2

Sum of the squares of the first n natural numbers ie. 12 + 23 + 32 + 42 + …+ n2 =  n n 1 2n 1 / 6

Types of Numbers:

  1. Natural Numbers – n > 0 where n is counting number; [1,2,3…]
  2. Whole Numbers – n ≥ 0 where n is counting number; [0,1,2,3…].

0 is the only whole number which is not a natural number.

Every natural number is a whole number.

  1. Integers – n ≥ 0 or n ≤ 0 where n is counting number;…,-3,-2,-1,0,1,2,3… are integers.

Positive Integers – n > 0; [1,2,3…]

Negative Integers – n < 0; [-1,-2,-3…]

Non-Positive Integers – n ≤ 0; [0,-1,-2,-3…]

Non-Negative Integers – n ≥ 0; [0,1,2,3…]

0 is neither positive nor negative integer.

  1. Even Numbers – n / 2 = 0 where n is counting number; [0,2,4,…]
  2. Odd Numbers – n / 2 ≠ 0 where n is counting number; [1,3,5,…]
  3. Prime Numbers – Numbers which is divisible by themselves only apart from 1.

1 is not a prime number.

To test a number p to be prime, find a whole number k such that k > √p. Get all prime numbers less than or equal to k and divide p with each of these prime numbers. If no number divides p exactly then p is a prime number otherwise it is not a prime number.

Example: 191 is prime number or not?

Solution:

Step 1 – 14 > √191

Step 2 – Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 – 191 is not divisible by any above prime number.

Result – 191 is a prime number.

Example: 187 is prime number or not?

Solution:

Step 1 – 14 > √187

Step 2 – Prime numbers less than 14 are 2,3,5,7,11 and 13.

Step 3 – 187 is divisible by 11.

Result – 187 is not a prime number.

  1. Composite Numbers – Non-prime numbers > 1. For example, 4,6,8,9 etc.
  • 1 is neither a prime number nor a composite number.
  • 2 is the only even prime number.
  1. Co-Primes Numbers – Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.

Rules for Divisibility

Divisibility by 2: A number is divisible by 2 when the digit at ones place is 0, 2, 4, 6 or 8.

 eg, 3582, 460, 28, 352, ….

Divisibility by 3: A number is divisible by 3 when sum of all digits of a number is a multiple of 3.

 eg, 453 = 4 + 5 + 3 = 12.

12 is divisible by 3 so, 453 is also divisible by 3.

Divisibility by 4: A number is divisible by 4, if the number formed with its last two digits is divisible by 4. eg, if we take the number 45024, the last two digits form 24. Since, the number 24 is divisible by 4, the number 45024 is also divisible by 4.

Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.

   eg, 10, 25, 60

Divisibility by 6: A number is divisible by 6, if it is divisible both by 2 and 3.

   eg, 48, 24, 108

Divisibility by 7: A number is divisible by 7 when the difference between twice the digit at ones place and the number formed by other digits is either zero or a multiple of 7

.

  eg, 658

   65 – 2 × 8 = 65 – 16 = 49

As 49 is divisible by 7 the number 658 is also divisible by 7.

Divisibility by 8: A number is divisible by 8, if the number formed by the last 3 digits of the number is divisible by 8. eg, if we take the number 57832, the last three digits form 832. Since, the number 832 is divisible

by 8, the number 57832 is also divisible by 8..

Divisibility by 9: A number is divisible by 9, if the sum of all the digits of a number is a multiple of 9.

   eg, 684 = 6 + 8 + 4 = 18.

18 is divisible by 9 so, 684 is also divisible by 9.

Divisibility by 10: A number is divisible by 10, if its last digit is 0. eg, 20, 180, 350,….

Divisibility by 11: When the difference between the sum of its digits in odd places and in even places is either 0 or a multiple of 11.

  eg, 30426

  3 + 4 + 6 = 13

  0 + 2 = 2

  13 – 2 = 11

As the difference is a multiple of 11 the number 30426 is also divisible by 11.

Example 2:

 In a sum of division, the quotient is 110, the remainder is 250, the divisor is equal to the sum of the quotient and remainder. What is the dividend ?

Solution.

Divisor = (110 + 250) = 360
Dividend = (360 × 110) + 250 = 39850
Hence, the dividend is 39850.

Example 3: 

Find the number of numbers upto 600 which are divisible by 14. 

Solution.

Divide 600 by 13, the quotient obtained is 46. Thus, there are 46 numbers less than 600 which are divisible by 14.


Division on Numbers

In a sum of division, we have four quantities.
They are (i) Dividend, (ii) Divisor, (iii) Quotient and (iv) Remainder. These quantities are connected by a relation.

(a) Dividend = Divisor × Quotient + Remainder.
(b) Divisor = (Dividend – Remainder) ÷ Quotient.
(c) Quotient = (Dividend – Remainder) – Divisor.


Factors and Multiples

Factor: A number which divides a given number exactly is called a factor of the given number, 

eg, 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6

Thus, 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.

  • 1 is a factor of every number
  • A number is a factor of itself
  • The smallest factor of a given number is 1 and the greatest factor is the number itself.
  •  If a number is divided by any of its factors, the remainder is always zero.
  •  Every factor of a number is either less than or at the most equal to the given number.
  •  Number of factors of a number are finite.

Number of Factors of a Number

If N is a composite number such that N = am bn co… where a, b, c … are prime factors of N and m, n, o … are positive integers, then the number of factors of N is given by the expression (m + 1) (n + 1) (o + 1)

Example 4:

Find the number of factors that 224 has.

Solution.

224 = 25 × 71

Hence, 224 has (5 + 1) (1 + 1) = 6 × 2 = 12 factors.


Multiple

A multiple of a number is a number obtained by multiplying it by a natural number

eg,

Multiples of 5 are 5, 10, 15, 20
Multiples of 12 are 12, 24, 36, 48

  •  Every number is a multiple of 1.The smallest multiple of a number is the number itself.
  • We cannot find the greatest multiple of a number.
  • Number of multiples of a number are infinite.
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