Aptitude – Number system

Number system

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 321, 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 321:

• 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 10² = 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…] number system

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 number system

Following are tips to check divisibility of numbers. number system

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. number system

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.

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.

8. 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.

9. 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 number system

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. number system

Division Algorithm

When a number is divided by another number then

Dividend = (Divisor x Quotient) + Reminder

Series

Following are formulaes for basic number series:

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 Formulas

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)