Number System Questions and Concepts for Aptitude Tests

Q: How can I find the number of factors of a number?

First express the number as a product of prime powers. If N = p^a × q^b, then the number of positive factors is (a + 1)(b + 1).

Number System is one of the basic topics in quantitative aptitude. It helps you understand how numbers behave and makes many other aptitude topics easier to solve.

Questions from Number System can involve natural numbers, whole numbers, integers, prime numbers, factors, multiples, divisibility, remainders, HCF, LCM, unit digits, and place value.

For placement tests and competitive exams, you do not need to memorise every possible rule. You need to understand the basic properties of numbers and know when to apply them.

This guide explains the important Number System concepts with simple examples, formulas, shortcuts, common mistakes, and practice questions.

Quick Answer

A number system is a way of classifying and working with different types of numbers.

The most important number types for aptitude are:

Natural numbers
Whole numbers
Integers
Rational numbers
Irrational numbers
Prime numbers
Composite numbers
Even numbers
Odd numbers

Some of the most useful Number System concepts are:

Factors and multiples
Divisibility rules
Prime factorisation
HCF and LCM
Remainders
Unit digits
Number of factors

Pocket Aptitude Tip: Do not try to memorise Number System as a collection of unrelated rules. Understand how numbers are connected. The same concepts appear repeatedly in aptitude questions.

What is a Number System?

A number system is a method of representing and classifying numbers.

For aptitude preparation, the first step is to understand the different types of numbers.

Natural Numbers

Natural numbers are the counting numbers:

1, 2, 3, 4, 5, ...

The smallest natural number is 1.

Whole Numbers

Whole numbers include zero along with the natural numbers:

0, 1, 2, 3, 4, 5, ...

So:

Whole numbers = Natural numbers + 0

Integers

Integers include positive numbers, negative numbers, and zero:

..., −3, −2, −1, 0, 1, 2, 3, ...

Rational Numbers

A rational number can be written in the form:

p/q

where p and q are integers and q ≠ 0.

Examples:

1/2
3/4
5
−7/3

Every integer is also a rational number because an integer can be written over 1.

For example:

5 = 5/1

Irrational Numbers

Irrational numbers cannot be written as a fraction of two integers.

Examples include:

√2
√3
π

Their decimal representations do not terminate or repeat in a fixed pattern.

Types of Numbers at a Glance

Type	Examples
Natural numbers	1, 2, 3, 4, …
Whole numbers	0, 1, 2, 3, …
Integers	…, −2, −1, 0, 1, 2, …
Rational numbers	1/2, 3/4, 5
Irrational numbers	√2, √3, π
Prime numbers	2, 3, 5, 7, 11
Composite numbers	4, 6, 8, 9, 10
Even numbers	2, 4, 6, 8, …
Odd numbers	1, 3, 5, 7, …

Prime Numbers

A prime number is a positive integer greater than 1 that has exactly two positive factors: 1 and itself.

Examples:

2, 3, 5, 7, 11, 13, 17, 19

For example, 7 has only two factors:

1 × 7

Therefore, 7 is prime.

Important Point

1 is neither prime nor composite. This is a common exam mistake.

Is 2 Prime?

Yes. 2 is the only even prime number. Every other even number is divisible by 2 and therefore has more than two factors.

Composite Numbers

A composite number is a positive integer greater than 1 that has more than two factors.

Examples:

4, 6, 8, 9, 10, 12

For example, the factors of 12 are:

1, 2, 3, 4, 6, 12

Since 12 has more than two factors, it is composite.

Even and Odd Numbers

An even number is divisible by 2.

Examples:

2, 4, 6, 8, 10, 12

An odd number is not divisible by 2.

Examples:

1, 3, 5, 7, 9, 11

Important Rules

Operation	Result
Even + Even	Even
Even + Odd	Odd
Odd + Odd	Even
Even × Even	Even
Even × Odd	Even
Odd × Odd	Odd

These rules can help eliminate options quickly in aptitude questions.

Factors and Multiples

Factors and multiples are closely related, but they are not the same thing.

Factors

A factor divides a number exactly.

For example, the factors of 12 are:

1, 2, 3, 4, 6, 12

because each of these numbers divides 12 without leaving a remainder.

Multiples

Multiples are obtained by multiplying a number by positive integers.

Multiples of 5 are:

5, 10, 15, 20, 25, ...

Factor vs Multiple

Factor	Multiple
Divides a number exactly	Obtained by multiplying
Usually smaller than or equal to the number	Usually greater than or equal to the number
3 is a factor of 12	12 is a multiple of 3

Common Mistake: Students often reverse the terms. If 4 divides 20 exactly, then 4 is a factor of 20, while 20 is a multiple of 4.

Divisibility Rules

Divisibility rules help you determine whether a number is divisible by another number without performing a complete division.

These rules are especially useful in aptitude tests.

Divisibility by 2

A number is divisible by 2 if its last digit is even.

Last digit can be: 0, 2, 4, 6, 8

Example: 348 — the last digit is 8. Therefore, 348 is divisible by 2.

Divisibility by 3

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

Example: 123

Digit sum: 1 + 2 + 3 = 6

Since 6 is divisible by 3, 123 is divisible by 3.

Divisibility by 4

A number is divisible by 4 if its last two digits form a number divisible by 4.

Example: 1,236 — last two digits: 36. Since 36 is divisible by 4, 1,236 is divisible by 4.

Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5.

Examples: 125, 340, 1,005

Divisibility by 6

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

Example: 132 — it is even, so it is divisible by 2. Digit sum: 1 + 3 + 2 = 6, so it is divisible by 3. Therefore, 132 is divisible by 6.

Divisibility by 8

A number is divisible by 8 if its last three digits form a number divisible by 8.

Example: 1,224 — last three digits: 224. Since 224 ÷ 8 = 28, the number is divisible by 8.

Divisibility by 9

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

Example: 729

Digit sum: 7 + 2 + 9 = 18

Since 18 is divisible by 9, 729 is divisible by 9.

Divisibility by 10

A number is divisible by 10 if its last digit is 0.

Example: 450 is divisible by 10.

Divisibility by 11

For divisibility by 11, find the difference between the sums of alternate digits.

Example: 121

Alternate digit sums: 1 + 1 = 2 and 2
Difference: 2 − 2 = 0

Since 0 is divisible by 11, 121 is divisible by 11.

Divisibility Rules Table

Number	Divisibility Rule
2	Last digit is even
3	Sum of digits is divisible by 3
4	Last two digits are divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
8	Last three digits are divisible by 8
9	Sum of digits is divisible by 9
10	Last digit is 0
11	Difference between alternate digit sums is divisible by 11

Speed Improvement Tip: Divisibility rules are most useful when a question asks you to identify a missing digit, test a large number, or eliminate answer choices quickly.

Prime Factorisation

Prime factorisation means expressing a number as a product of prime numbers.

For example:

60 = 2 × 2 × 3 × 5
or
60 = 2² × 3 × 5

Prime factorisation is useful for finding:

HCF
LCM
Number of factors
Common factors
Divisibility

Example

Find the prime factorisation of 72.

72 = 2 × 36
36 = 2 × 18
18 = 2 × 9
9 = 3 × 3

Therefore:

72 = 2³ × 3²

HCF and LCM

HCF means Highest Common Factor. LCM means Least Common Multiple.

These concepts are frequently used in aptitude questions.

HCF

The HCF of two or more numbers is the greatest number that divides all of them exactly.

Example: Find the HCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6

Therefore: HCF = 6

LCM

The LCM is the smallest positive number that is a multiple of all the given numbers.

Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 6: 6, 12, 18, 24, ...

The first common multiple is 12.

Therefore: LCM = 12

HCF and LCM Using Prime Factorisation

Consider:

24 = 2³ × 3
36 = 2² × 3²

HCF

For HCF, take the common prime factors with the smallest powers.

HCF = 2² × 3 = 12

LCM

For LCM, take all prime factors with the largest powers.

LCM = 2³ × 3² = 72

Important HCF-LCM Relationship

For two positive integers:

HCF × LCM = Product of the two numbers

For example, for 12 and 18:

HCF = 6
LCM = 36
6 × 36 = 216
12 × 18 = 216

So the relationship works.

Important: This direct product relationship is for two numbers. Do not blindly apply it to three or more numbers.

Remainders

A remainder is the amount left after division.

For example:

17 ÷ 5
5 × 3 = 15
17 − 15 = 2

Therefore, 17 ÷ 5 gives remainder 2.

The basic division relationship is:

Dividend = Divisor × Quotient + Remainder
17 = 5 × 3 + 2

Important Property

The remainder is always:

0 ≤ Remainder < Divisor

For example, when dividing by 7, a remainder can be 0, 1, 2, 3, 4, 5, or 6. It cannot be 7 or greater.

Remainder When a Number is Divided by Another Number

Suppose we need the remainder when 125 is divided by 7.

7 × 17 = 119
125 − 119 = 6

So the remainder is 6.

Remainder of a Power

Questions involving large powers can look difficult, but patterns often make them easier.

Consider powers of 2:

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32

If we are interested in the last digit, the pattern is:

2, 4, 8, 6

Then it repeats:

2, 4, 8, 6, 2, 4, 8, 6...

This repeating pattern can help solve questions involving very large powers.

Unit Digit

The unit digit is the last digit of a number.

For example:

Unit digit of 347 = 7
Unit digit of 5,892 = 2
Unit digit of 1,000 = 0

Unit-digit questions often involve powers.

Example

Find the unit digit of 2⁵.

2⁵ = 32

Therefore, unit digit = 2

For larger powers, identify the repeating pattern.

Unit Digit Patterns

Some common powers have repeating unit-digit cycles.

Number	Repeating Unit Digits
2	2, 4, 8, 6
3	3, 9, 7, 1
4	4, 6
7	7, 9, 3, 1
8	8, 4, 2, 6
9	9, 1

For numbers ending in:

0 → unit digit always 0
1 → unit digit always 1
5 → unit digit always 5
6 → unit digit always 6

Example

Find the unit digit of 3⁷.

The pattern for powers of 3 is:

3, 9, 7, 1

The cycle length is 4. Since 7 ÷ 4 gives remainder 3, we use the third number in the cycle: 7.

Therefore, unit digit = 7

Number of Factors

If a number is written in prime factorised form:

N = pᵃ × qᵇ × rᶜ

then the number of positive factors is:

(a + 1)(b + 1)(c + 1)

Example: Number of Factors of 60

60 = 2² × 3 × 5
Number of factors = (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12

Therefore, 60 has 12 positive factors.

Example: Number of Factors of 72

72 = 2³ × 3²
Number of factors = (3 + 1)(2 + 1) = 4 × 3 = 12

So 72 has 12 positive factors.

Perfect Squares and Number of Factors

A perfect square has an important factor property.

If a number is a perfect square, its number of positive factors is odd.

For example:

36 = 2² × 3²
Number of factors = (2 + 1)(2 + 1) = 9

So 36 has 9 factors, which is odd.

Why? Most factors occur in pairs:

The middle factor, 6, is paired with itself. That is why perfect squares have an odd number of factors.

Common Mistakes in Number System

1. Treating 1 as a Prime Number

1 is neither prime nor composite.

2. Confusing Factors and Multiples

If 5 divides 20, 5 is a factor of 20 and 20 is a multiple of 5.

3. Forgetting that 2 is the Only Even Prime

2 is prime. 4, 6, 8 and other even numbers greater than 2 are composite.

4. Using the Wrong Divisibility Rule

For example, divisibility by 4 depends on the last two digits, not the sum of all digits.

5. Ignoring the Remainder Condition

When dividing by n, the remainder must be less than n.

6. Using HCF and LCM Incorrectly

For prime factorisation, HCF uses the smallest powers of common prime factors, and LCM uses the largest powers of all required prime factors.

7. Calculating Large Powers Directly

When the question asks for a unit digit or remainder, look for a repeating pattern first.

Exam Tip: Many Number System questions are designed to test whether you recognise a pattern. Long calculations are often unnecessary.

Fast Approach for Number System Questions

When you see a Number System question, first identify what the question is asking.

If it asks about divisibility — check the appropriate divisibility rule.

If it asks about HCF or LCM — try prime factorisation.

If it asks about factors — prime-factorise the number and use the factor-count formula.

If it asks about a unit digit — look for the repeating pattern of the last digit.

If it asks about a remainder — write the division relationship:

Dividend = Divisor × Quotient + Remainder

Then simplify the expression before performing large calculations.

If it asks whether a number is prime — check whether it has any divisor other than 1 and itself. For a number n, you only need to test possible prime divisors up to √n to determine whether it is prime.

Pocket Aptitude Speed Tip: Before calculating, ask: “What property of numbers can solve this?” This simple habit can save significant time.

Practice Questions

Before you calculate, take a guess. Pick an option for each question below to reveal the full solution — your score tallies up as you go.

0 / 12 correct

Question 1Easy

Which of the following is neither prime nor composite?