Number system - Aptitude Questions & Answers
Number system is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.
Mastering the number system is fundamental for aptitude tests and competitive exams. This comprehensive guide covers everything from basic number classification to advanced concepts like unit digits and remainder theorems, explained with clear examples and practical applications.
Number System: Complete Guide with Examples & Formulas
What is a Number?
A number is an arithmetic value used to represent quantity in mathematics. Understanding numbers forms the foundation for all mathematical operations and problem-solving in aptitude tests.
Classification of Numbers
Numbers are systematically organized into different categories based on their properties:
Complex Numbers
Complex numbers have both a real and an imaginary part, typically written as (a + ib), where a and b are real numbers and i is the imaginary unit (iΒ² = -1).
Example: (2 + 3i), (32 + 5i), etc.
Real Numbers
Any number that can be found on the number line, including both rational and irrational numbers.
Examples: 7.4, 12.1, 23, 45/7, β2, β5, 3.565574...
Rational Numbers
Numbers that can be expressed as p/q where p and q are integers and q β 0.
Examples: β , β , 1.9, 5, 89
Further classification: Integer and Fraction
Irrational Numbers
Numbers that cannot be expressed as p/q where p and q are integers. These are non-terminating and non-repeating decimals.
Examples: β2, β3, Ο, 4.56746..., 985.4674...
Integers
Include all positive and negative natural numbers including zero.
Examples: -5, -6, 7, 8, 0, etc.
Whole Numbers
The set of positive integers including zero.
Examples: 0, 1, 2, 3, 4, 5, etc.
Natural Numbers
The set of positive integers, excluding zero.
Examples: 1, 3, 5, 6, 8, etc.
Fractions
Numbers expressed as p/q where p and q are natural numbers.
Example: β (3 = numerator, 5 = denominator)
Types: Proper, Improper, and Mixed Fractions
Types of Fractions
Proper Fraction
Numerator < Denominator. The part is less than the whole.
Example: β (2 < 3)
Improper Fraction
Numerator β₯ Denominator. The part is equal to or more than the whole.
Example: 7/4 (7 > 4)
Mixed Fraction
Combines a natural number and a proper fraction.
Example: 2ΒΌ (2 + ΒΌ)
Divisibility Rules for Quick Calculations
Divisibility by 2
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
Example: 26 is divisible by 2 because last digit is 6.
Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: 369 (3+6+9=18, 18Γ·3=6)
Divisibility by 4
A number is divisible by 4 if the last two digits form a number divisible by 4.
Example: 132 (32Γ·4=8)
Divisibility by 5
A number is divisible by 5 if its last digit is 0 or 5.
Example: 725 (last digit 5)
Divisibility by 6
A number is divisible by 6 if it\'s divisible by both 2 and 3.
Example: 294 (even + digit sum 15Γ·3=5)
Divisibility by 7
Double the last digit, subtract from remaining number. If result is 0 or divisible by 7.
Example: 798 β 79 - (8Γ2=16) = 63 β 63Γ·7=9 β
Divisibility by 8
A number is divisible by 8 if the last three digits form a number divisible by 8.
Example: 2416 (416Γ·8=52)
Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example: 2,763 (2+7+6+3=18, 18Γ·9=2)
Divisibility by 10
A number is divisible by 10 if it ends in 0.
Example: 430 (last digit 0)
Divisibility by 11
Difference between sum of odd and even position digits should be 0 or divisible by 11.
Example: 292215
(9+2+5) - (2+2+1) = 16 - 5 = 11 β
Divisibility by 12
A number is divisible by 12 if it\'s divisible by both 4 and 3.
Example: 864 (divisible by 4 & 3)
Co-Prime Numbers
Co-prime numbers are pairs of numbers that don\'t share any common factors except 1. Their HCF is always 1, and LCM equals their product.
Example: 5 and 6 are co-prime (only common factor is 1)
Key Properties:
- HCF = 1
- LCM = Product of numbers
Common Co-prime pairs: (1,2), (2,3), (3,4), (3,5), etc.
Division Algorithm Formula
Where 0 β€ r < b
Where:
- a = Dividend
- b = Divisor
- n = Quotient
- r = Remainder
Example: 26 = 8 Γ 3 + 2
Unit Digit Concept: Power Cycles & Patterns
What is Unit Digit?
The rightmost digit of a number (ones place). Crucial for solving power and multiplication problems quickly.
Example: Unit digit of 847 is 7
Two Types of Unit Digit Problems
1. Simple Product Type
Multiply only the unit digits of numbers.
Example: 478 Γ 593
Unit digits: 8 Γ 3 = 24 β Unit digit is 4
2. Power Cycle Type
Recognize repeating patterns when numbers are raised to powers.
Category 1: Digits 0, 1, 5, 6
Any power (except 0) gives the same unit digit as the number itself.
5Β²=25, 1β΅=1, 0βΉ=0, 6Β²=36
Category 2: Digits 4 and 9
Cyclicity of 2 patterns based on odd/even powers.
Pattern for 4:
- 4odd = 4
- 4even = 6
Pattern for 9:
- 9odd = 9
- 9even = 1
Example: 3456 β 56 is even β Unit digit = 6
Category 3: Digits 2, 3, 7, 8
Cyclicity of 4 different numbers.
2: 2, 4, 8, 6
3: 3, 9, 7, 1
7: 7, 9, 3, 1
8: 8, 4, 2, 6
Example: 253125
125 Γ· 4 = remainder 1 β Cycle position 1 β Unit digit = 3
Trailing Zeroes in Factorials & Products
Trailing zeros are zeros at the end of a number, created by factors of 10 (2Γ5).
Step-by-Step Method:
- Prime Factorization: Break number into prime factors
- Count 2s and 5s: Note the number of 2 and 5 factors
- Calculate Trailing Zeros: Minimum(count of 2s, count of 5s)
Example: 4 Γ 9 Γ 125
Prime factors: 2Β² Γ 3Β² Γ 5Β³
Count of 2s = 2, Count of 5s = 3
Minimum = 2 β 2 trailing zeros
Remainder Theorem: Essential Rules & Applications
What is Remainder?
Where 0 β€ R < x
Example: 10 Γ· 3 = 3 remainder 1
10 = 3 Γ 3 + 1
Basic Remainder Theorem
When dividing a product of numbers, find individual remainders first, then multiply them.
Formula:
(a Γ b Γ c) Γ· d = Multiply individual remainders, then divide by d
Example: (9 Γ 8 Γ 6) Γ· 5
- 9Γ·5 β remainder 4
- 8Γ·5 β remainder 3
- 6Γ·5 β remainder 1
- Multiply: 4Γ3Γ1 = 12
- 12Γ·5 β remainder 2
Exponential Remainder Cycles Theorem
For large exponents, find repeating remainder patterns (cycles).
Example 1: 2100 Γ· 5
Cycle for 2 modulo 5: 2, 4, 3, 1 (repeats every 4)
100 Γ· 4 = remainder 0 β position 4 in cycle β remainder = 1
Example 2: 423 Γ· 12
All powers of 4 give remainder 4 when divided by 12
Remainder = 4
Example 3: 17240 Γ· 7
- 17 mod 7 = 3 β Find 3240 mod 7
- Cycle for 3 modulo 7: 3, 2, 6, 4, 5, 1 (length 6)
- 240 Γ· 6 = remainder 0 β position 6 in cycle β remainder = 1
Frequently Asked Questions
What is the difference between rational and irrational numbers?
Rational numbers can be expressed as fractions (p/q where qβ 0), while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions.
How to quickly check divisibility by 7?
Double the last digit, subtract from the remaining number. If result is 0 or divisible by 7, original number is divisible by 7.
Why is the unit digit concept important?
It helps solve complex multiplication and exponent problems quickly without full calculation, saving time in competitive exams.
What are co-prime numbers used for?
Co-prime numbers simplify fractions, help in finding HCF/LCM, and are fundamental in number theory and cryptography.
Frequently Asked Questions
What is Number system?
Number system is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.
Is Number system important for competitive exams?
Yes, Number system is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.
How to prepare Number system easily?
Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Number system.
What are the important formulas in Number system?
Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Number system.
How many questions come from Number system?
Typically 5-10 questions come from Number system in most competitive exams, making it a high-scoring section.
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