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Number System
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Number System
The Number System is an important topic in the Quantitative Aptitude section of most placement tests and competitive exams. It includes different types of numbers, their properties, and operations.
A strong understanding of the number system helps candidates solve problems related to:
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Divisibility
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Factors and multiples
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LCM and HCF
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Number series
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Basic arithmetic operations
Types of Numbers
Natural Numbers
The set of positive integers starting from 1.
Example:
1, 2, 3, 4, 5, …
Whole Numbers
Natural numbers along with zero.
Example:
0, 1, 2, 3, 4, …
Integers
All positive and negative whole numbers, including zero.
Example:
…, −3, −2, −1, 0, 1, 2, 3, …
Rational Numbers
Numbers that can be written in the form p/q, where:
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p and q are integers
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q ≠ 0
Includes:
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Fractions
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Terminating decimals
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Repeating decimals
Example:
1/2, 3/4, 0.25, 0.333…
Irrational Numbers
Numbers that cannot be expressed as fractions and have non-terminating, non-repeating decimals.
Example:
√2, √3, π
Real Numbers
The set of all rational and irrational numbers.
All real numbers can be represented on the number line.
Factors and Multiples
Factors
Factors are numbers that divide another number without leaving a remainder.
Example:
Factors of 12:
1, 2, 3, 4, 6, 12
Key Points:
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Every number has at least two factors: 1 and itself.
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A prime number has exactly two factors.
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Factors are always integers.
Multiples
Multiples are numbers obtained by multiplying a given number by integers.
Example:
Multiples of 15:
15, 30, 45, 60, …
Key Points:
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Every number is a multiple of itself.
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Multiples are infinite.
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Multiples are always integers.
Relationship Between Factors and Multiples
If X is a factor of Y, then Y is a multiple of X.
Example:
4 is a factor of 16
So, 16 is a multiple of 4.
Prime and Composite Numbers
Prime Numbers
Numbers greater than 1 with exactly two factors: 1 and the number itself.
Example:
2, 3, 5, 7, 11, 13
Composite Numbers
Numbers that have more than two factors.
Example:
4, 6, 8, 9, 10
LCM and HCF (GCD)
LCM (Least Common Multiple)
The smallest number that is a multiple of two or more numbers.
Example:
LCM of 15 and 25:
Multiples of 15: 15, 30, 45, 60, 75
Multiples of 25: 25, 50, 75
LCM = 75
HCF or GCD (Greatest Common Divisor)
The largest number that divides two or more numbers exactly.
Example:
HCF of 15 and 25:
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
HCF = 5
Divisibility Rules
These rules help determine whether a number is divisible by another number without performing full division.
Divisibility by 2
A number is divisible by 2 if its last digit is even.
Example:
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426 → divisible by 2
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315 → not divisible by 2
Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
738 → 7 + 3 + 8 = 18
18 is divisible by 3
So, 738 is divisible by 3.
Number Series
Number series questions require identifying a pattern and finding the missing or next number.
Arithmetic Series
Each term increases by a constant difference.
Example:
2, 5, 8, 11, 14, …
Common difference = 3
Geometric Series
Each term is multiplied by a constant ratio.
Example:
2, 6, 18, 54, 162, …
Common ratio = 3
Fibonacci Series
Each term is the sum of the two previous terms.
Example:
0, 1, 1, 2, 3, 5, 8, 13, …
Square Number Series
Each term is the square of an integer.
Example:
1, 4, 9, 16, 25, 36, …
Cube Number Series
Each term is the cube of an integer.
Example:
1, 8, 27, 64, 125, …
Prime Number Series
A sequence of prime numbers.
Example:
2, 3, 5, 7, 11, 13, 17, …
Number Properties
Common properties tested in aptitude exams:
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Even and odd numbers
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Perfect squares
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Perfect cubes
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Divisibility properties
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Remainder concepts
Tips for Solving Number System Questions
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Memorise divisibility rules.
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Learn prime numbers up to at least 50.
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Practise LCM and HCF problems.
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Identify patterns quickly in number series.
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Break complex problems into smaller steps.
