Aptitude - Number System

Numbers

In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals.

Face Value

Face value of a digit in a numeral is value of the digit itself. For example in 123, face value of 1 is 1, face value of 2 is 2 and face value of 3 is 3.

Place Value

Place value of a digit in a numeral is value of the digit multiplied by 10ⁿ where n starts from 0. For example in 123:

• Place value of 1 = 1 x 10⁰ = 1 x 1 = 1

• Place value of 2 = 2 x 10¹ = 2 x 10 = 20

• Place value of 3 = 3 x 10¹ = 3 x 100 = 300

0^th position digit is called unit digit and is the most commonly used topic in aptitude tests.

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.

3. 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.

4. Even Numbers - n / 2 = 0 where n is counting number; [0,2,4,...]

5. Odd Numbers - n / 2 ≠ 0 where n is counting number; [1,3,5,...]

6. 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.  

7. 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.

8. 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.

Divisibility

Following are tips to check divisibility of numbers.

1. Divisibility by 2 - A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.

Example: 64578 is divisible by 2 or not?  Solution:   Step 1 - Unit digit is 8.  Result - 64578 is divisible by 2.    Example: 64575 is divisible by 2 or not?  Solution:   Step 1 - Unit digit is 5.  Result - 64575 is not divisible by 2.  

2. Divisibility by 3 - A number is divisible by 3 if sum of its digits is completely divisible by 3.

Example: 64578 is divisible by 3 or not?  Solution:   Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30   which is divisible by 3.  Result - 64578 is divisible by 3.    Example: 64576 is divisible by 3 or not?  Solution:   Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28   which is not divisible by 3.  Result - 64576 is not divisible by 3.  

3. Divisibility by 4 - A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.

Example: 64578 is divisible by 4 or not?  Solution:   Step 1 - number formed using its last two digits is 78   which is not divisible by 4.  Result - 64578 is not divisible by 4.    Example: 64580 is divisible by 4 or not?  Solution:   Step 1 - number formed using its last two digits is 80   which is divisible by 4.  Result - 64580 is divisible by 4.  

4. Divisibility by 5 - A number is divisible by 5 if its unit digit is 0 or 5.

Example: 64578 is divisible by 5 or not?  Solution:   Step 1 - Unit digit is 8.  Result - 64578 is not divisible by 5.    Example: 64575 is divisible by 5 or not?  Solution:   Step 1 - Unit digit is 5.  Result - 64575 is divisible by 5.  

5. Divisibility by 6 - A number is divisible by 6 if the number is divisible by both 2 and 3.

Example: 64578 is divisible by 6 or not?  Solution:   Step 1 - Unit digit is 8. Number is divisible by 2.  Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 8 = 30   which is divisible by 3.  Result - 64578 is divisible by 6.    Example: 64576 is divisible by 6 or not?  Solution:   Step 1 - Unit digit is 8. Number is divisible by 2.  Step 2 - Sum of its digits is 6 + 4 + 5 + 7 + 6 = 28   which is not divisible by 3.  Result - 64576 is not divisible by 6.  

6. Divisibility by 8 - A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.

Example: 64578 is divisible by 8 or not?  Solution:   Step 1 - number formed using its last three digits is 578   which is not divisible by 8.  Result - 64578 is not divisible by 8.    Example: 64576 is divisible by 8 or not?  Solution:   Step 1 - number formed using its last three digits is 576   which is divisible by 8.  Result - 64576 is divisible by 8.  

7. Divisibility by 9 - A number is divisible by 9 if sum of its digits is completely divisible by 9.

8. Divisibility by 10 - A number is divisible by 10 if

Example: 64579 is divisible by 9 or not?  Solution:   Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 9 = 31   which is not divisible by 9.  Result - 64579 is not divisible by 9.    Example: 64575 is divisible by 9 or not?  Solution:   Step 1 - Sum of its digits is 6 + 4 + 5 + 7 + 5 = 27   which is divisible by 9.  Result - 64575 is divisible by 9.  

9. Divisibility by 10 - A number is divisible by 10 if its unit digit is 0.

Example: 64575 is divisible by 10 or not?  Solution:   Step 1 - Unit digit is 5.  Result - 64578 is not divisible by 10.    Example: 64570 is divisible by 10 or not?  Solution:   Step 1 - Unit digit is 0.  Result - 64570 is divisible by 10.  

10. Divisibility by 11 - A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.

Example: 64575 is divisible by 11 or not?  Solution:   Step 1 - difference between sum of digits at odd places   and sum of digits at even places = (6+5+5) - (4+7) = 5   which is not divisible by 11.  Result - 64575 is not divisible by 11.    Example: 64075 is divisible by 11 or not?  Solution:   Step 1 - difference between sum of digits at odd places   and sum of digits at even places = (6+0+5) - (4+7) = 0.  Result - 64075 is divisible by 11.  

Tips on Division

1. If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.

2. (a-b) always divides (aⁿ - bⁿ) if n is a natural number.

3. (a+b) always divides (aⁿ - bⁿ) if n is an even number.

4. (a+b) always divides (aⁿ + bⁿ) if n is an odd number.

Division Algorithm

When a number is divided by another number then

Dividend = (Divisor x Quotient) + Reminder

Series

Following are formulaes for basic number serires:

1. (1+2+3+...+n) = (1/2)n(n+1)

2. (1²+2²+3²+...+n²) = (1/6)n(n+1)(2n+1)

3. (1³+2³+3³+...+n³) = (1/4)n²(n+1)²

Basic Formulaes

These are the basic formulae:

(a + b)2 = a2 + b2 + 2ab
  (a - b)2 = a2 + b2 - 2ab
  (a + b)2 - (a - b)2 = 4ab
  (a + b)2 + (a - b)2 = 2(a2 + b2)
  (a2 - b2) = (a + b)(a - b)
  (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
  (a3 + b3) = (a + b)(a2 -  ab + b2)
  (a3 - b3) = (a - b)(a2 + ab + b2)
  (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)