Numerical Ability -Number System and Simplification for Competitive Exams

Number System & Simplification

Classification of Numbers:

Odd Numbers:

Numbers which are not divisible by 2

Ex: 1, 3, 5, 7, ........

Even Numbers:

Numbers which are divisible by 2

Ex: 2, 4, 6, 8, .......

Natural Numbers (N):

All counting numbers

Ex: N = {1, 2, 3, ...... }

Sum of first 'n' natural numbers is  àn(n+1)/2

Sum of squares of first 'n' natural numbers is   à n(n + 1)(2n + 1)/6

Sum of cubes of first 'n' natural numbers  à (n(n + 1)/2)²

Whole Numbers (W):

0 and all natural numbers

Ex: W = { 0, 1, 2, 3, ... }

Integers (Z):

Natural numbers, zero and negative of natural numbers

Ex: I = { ......, −3, −2, −1, 0, +1, +2, +3, ...}

Rational Numbers (Q)

These are of the form of the form of p/q, where  q ≠ 0

Ex: Q = 1/5,2/3 ,0.-3 etc.

Irrational Numbers

Can't be written in p/q form, where q ≠ 0

Ex: √2, √5, 4 + √⎯7, π etc

Composite Numbers

Numbers having more than two factors

Prime Numbers

Numbers divisible by one and the number itself or numbers having only two factors.

Ex: 2, 3, 5, 7, 11, 13, ..... etc

Some points about prime numbers

★ All prime numbers end with 1, 3, 7 or 9 except 2 and 5

★ 2 is the only even prime number

★ '1' is neither a prime nor composite number. It is a unitary.

★ There are 25 prime numbers upto 100 and 15 prime numbers upto 50.

Twin prime numbers:

The difference between two prime numbers is '2'

Ex: 3 − 5, 5 − 7, 11 − 13 etc

Co prime numbers:

Numbers not having any common factors

Ex: 3, 5 and 8 are co primes to each other.

Prime Numbers:

	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

Divisibility Tests

1)      A number is divisible by 2 if the last digit is 0 or Even.

Ex:2758 is divisible by 2 since the last digit is 8.

2)      A number is divisible by 3 if the sum of the digits is divisible by 3.

Ex:168 is divisible by 3 since the sum of the digits is 15 (1 + 6 + 8 = 15) is divisible by 3.

3)      A number is divisible by 4 if the number formed by the last two digits is divisible by 4.

Ex: 7216 is divisible by 4 since 16 is divisible by 4.

4)      A number is divisible by 5 if the last digit is either 0 or 5.

Ex:: 3815 is divisible by 5 since the last digit is 5.

5)      A number is divisible by 6 if it is divisible by 2 and 3.

Ex:6198 is divisible by 6 since it is divisible by 2  AND it is divisible by 3.

6)      A number is divisible by 8 if the number formed by the last three digits is divisible by 8.

Ex:7152 is divisible by 8 since 152 is divisible by 8.

7)      A number is divisible by 9 if the sum of the digits is divisible by 9.

Ex:54279 is divisible by 9 since the sum of the digits 27 (5 + 4 + 2 + 7 + 9 = 27) is divisible by 9.

8)      A number is divisible by 10 if the last digit is 0.

Ex: 8370 is divisible by 10 since the last digit is 0.

9)      A number is divisible by 11 if the difference between the sum of digits of the odd places andeven  places is 0 or multiple of 11.

Ex: 3978295783 is divisible by 11 since difference of sum of odd place digits (3 + 7 + 2 + 5 + 8 = 25) and even place digits (9 + 8 + 9 + 7 + 3 = 36) is 11 (36 − 25 = 11)

Ex: Which of the following number is divisible by 9?

A) 37589457 B) 74655835 C) 89472519 D) 91632734 E) 83652927

Sol: Sum of the digits of the number 89472519 (8 + 9 + 4 + 7 + 2 + 5 + 1 + 9 = 45) is only divisible by 9 hence answer is (C)

Simplifications

In simplifying an expression the mathematical operations are performed in the following order.

V − Vinculum or Bar

B − Bracket

O − Of

D − Division

M − Multiplication

A − Addition

S − Subtraction

Ex: 64097 ÷ (44 × 0.25) + 3218 = ?

A) 9133 B) 9256 C) 9045

D) 67 E) None of these

To solve the equations based on numbers, the number is written in algebraic form.

For example Let the number be 'x' Then

--Three-fourth of a number = x ×3/4= 3x/4

--Two-fifth of a number = x ×2/5 =2x /5

--Double of a number = 2x,

--Thrice of a number = 3x

--30% of a number = 30/100 × x = 3x/10

--40% of 2/3  of one-fourth of a number = x ×1/4 ×2/3 ×40/100 = 2x/30

Laws Of Indices

1. a^m × aⁿ= a^m+n                       Ex: 7⁴ × 7³= 7⁴⁺³ = 7⁷

2. a^m/aⁿ= a^m−n              Ex: = 2⁹/2³= 2^{9 − 3} = 2⁶

3. (a^m)ⁿ= a^mn                Ex: (9³)²= 9^3×2

4. (ab)^m= a^m × b^m        Ex: (3 × 5)²= 3² × 5²

5. a^−m = 1/a^m                               Ex: 8⁻²= 1/8²= 1/64

6. m√a = a^1/m               Ex: 7√19 = 19^1/7

7. a⁰= 1                        Ex: 3⁰⁼= 1

Ex: 49⁸÷ 7⁵= (7)^?

Sol: (7²)⁸ /7³   à7¹⁶/7³=7^16-3=7¹³

∴ ? = 13