Unit digit

Concept of Unit Digit in Number System

The concept of unit digit is one of the most important topics in number system and quantitative aptitude. Questions based on unit digit are very common in competitive examinations like SSC, Banking, NDA, Railways, CAT, and other aptitude tests. These questions help test logical thinking, pattern recognition, and calculation speed.

By finding the unit digit of a number, we can often determine useful properties such as whether the number is even or odd, whether it can be divisible by certain numbers, and how powers of large numbers behave. This makes the unit digit concept a very scoring topic in aptitude preparation.

For example, if the unit digit of a number is $0, 2, 4, 6,$ or $8$, the number is even. If the unit digit is $1, 3, 5, 7,$ or $9$, the number is odd.

This simple idea becomes extremely powerful when solving powers, exponents, and simplification questions.

Concept of Cyclicity in Unit Digit Questions

It is very easy to identify the unit digit of a small number by direct observation. However, when a number is raised to a very large power such as $7^{103}$ or $9^{245}$, finding the unit digit becomes difficult using normal calculation.

This is where the concept of cyclicity becomes important.

Cyclicity is one of the most important shortcut methods used to solve unit digit questions quickly in competitive exams.

What is Cyclicity in Unit Digit?

The concept of cyclicity is based on the fact that every digit has its own repeating pattern of unit digits when raised to different powers.

This repeating sequence is called the cycle of that digit.

Instead of calculating the complete value of a large power, we only study the repeating pattern of the last digit.

This saves a lot of time in aptitude exams.

Example of Cyclicity for Digit 2

Let us observe the powers of $2$:

$2^1 = 2$ → unit digit = $2$

$2^2 = 4$ → unit digit = $4$

$2^3 = 8$ → unit digit = $8$

$2^4 = 16$ → unit digit = $6$

$2^5 = 32$ → unit digit = $2$

$2^6 = 64$ → unit digit = $4$

Here, the unit digits follow the sequence:

$2,\ 4,\ 8,\ 6$

After this, the same pattern repeats again.

This means the cyclicity of digit $2$ is $4$.

In simple words, after every $4$ powers, the unit digit repeats.

This is why finding the unit digit of large powers becomes easy.

Why Cyclicity is Important in Competitive Exams

Most unit digit questions in aptitude exams involve very large exponents.

For example:

Find the unit digit of $3^{57}$

Without cyclicity, solving this would take a long time.

With cyclicity, we simply identify the repeating pattern and use the remainder method.

This makes the question very fast and easy.

That is why cyclicity is one of the most important concepts in quantitative aptitude and number system.

Cyclicity of Other Digits

Every digit from $0$ to $9$ has its own cyclic pattern.

Some digits repeat after $4$ steps, some after $2$, and some remain the same always.

For example:

• digit $0$ always gives unit digit $0$

• digit $1$ always gives unit digit $1$

• digit $5$ always gives unit digit $5$

• digit $6$ always gives unit digit $6$

These digits have cyclicity of $1$

Digits like $2, 3, 7,$ and $8$ usually have cyclicity of $4$

Digits like $4$ and $9$ usually have cyclicity of $2$

Understanding these patterns helps solve unit digit questions much faster.

Cyclicity Table for Unit Digits

The cyclicity table is one of the most useful tools for solving unit digit questions quickly. It shows the repeating pattern of unit digits for all single-digit numbers.

How to Find the Unit Digit of a Number

Finding the unit digit of a number is one of the most important concepts in number system and quantitative aptitude. Unit digit questions are very common in competitive exams like SSC, Banking, NDA, Railways, CAT, and other aptitude tests because they can be solved quickly using logic instead of lengthy calculations.

There are mainly two types of problems when finding the unit digit of a number:

• Finding the unit digit of a number formed by multiplying several numbers

• Finding the unit digit of a number in the form of $x^n$

Both types follow different methods, so we need to understand them separately.

Unit Digit of a Number Formed by Multiplying Several Numbers

To find the unit digit of a number formed by multiplying many numbers, we only need to multiply the unit digits of those numbers.

There is no need to multiply the complete numbers.

This shortcut saves a lot of time in aptitude exams.

Rule for Finding Unit Digit in Multiplication

Step 1: Identify the unit digit of each number

Step 2: Multiply only the unit digits

Step 3: Take the unit digit of the final product

This gives the required answer.

Example

Find the unit digit of the number formed by:

$81 \times 82 \times 83 \times 84 \times 85 \times 86 \times 87 \times 88 \times 89$

Solution

Take only the unit digits:

$1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$

Now simplify step by step:

$1 \times 2 = 2$

$2 \times 3 = 6$

$6 \times 4 = 24$

Unit digit = $4$

Now:

$4 \times 5 = 20$

Unit digit = $0$

Once the unit digit becomes $0$, the final unit digit will always remain $0$

Therefore, the required unit digit is:

$0$

This is one of the fastest unit digit shortcut methods.

Unit Digit of a Number in the Form of $x^n$

To find the unit digit of a number in the form of $x^n$, we classify numbers based on their last digit.

Different ending digits follow different cyclic patterns.

These are divided into three important groups:

• Numbers ending with $0, 1, 5, 6$

• Numbers ending with $2, 3, 7, 8$

• Numbers ending with $4, 9$

This classification makes unit digit questions much easier.

Unit Digit of Numbers Ending with $0, 1, 5, 6$

If a number ends with $0, 1, 5,$ or $6$, the unit digit remains the same even after raising it to any power.

This means:

$0^n \Rightarrow 0$

$1^n \Rightarrow 1$

$5^n \Rightarrow 5$

$6^n \Rightarrow 6$

for any positive integer value of $n$

Important Rule

• The unit digit of a number ending with $0$ is always $0$

• The unit digit of a number ending with $1$ is always $1$

• The unit digit of a number ending with $5$ is always $5$

• The unit digit of a number ending with $6$ is always $6$

This is the easiest case in unit digit problems.

Example

Find the unit digit of:

$30^{24},\ 41^{36},\ 75^{44},\ 86^{38}$

Solution

$30^{24}$ ends with $0$

So, the unit digit is $0$

$41^{36}$ ends with $1$

So, the unit digit is $1$

$75^{44}$ ends with $5$

So, the unit digit is $5$

$86^{38}$ ends with $6$

So, the unit digit is $6$

These questions can be solved instantly without calculation.

Unit Digit of Numbers Ending with $2, 3, 7, 8$

These digits follow a cyclicity of $4$

This means their unit digits repeat after every $4$ powers.

To solve such questions, we use the remainder method.

Step-by-Step Method

Step 1: Identify the unit digit of the base number

Step 2: Divide the power $n$ by $4$

Step 3: Find the remainder

Possible remainders are:

$0,\ 1,\ 2,\ 3$

Step 4: Use the remainder to determine the final unit digit

This method is used for digits ending in $2, 3, 7,$ and $8$

Unit Digit of Numbers Ending with $2$

Pattern of powers of $2$:

$2^1 = 2$

$2^2 = 4$

$2^3 = 8$

$2^4 = 16$

Unit digit pattern:

$2,\ 4,\ 8,\ 6$

Rule for Digit $2$

If remainder is $1$ → unit digit = $2$

If remainder is $2$ → unit digit = $4$

If remainder is $3$ → unit digit = $8$

If remainder is $0$ → unit digit = $6$

Example

Find the unit digit of $4562^{75}$

Solution

The unit digit of $4562$ is $2$

Now divide:

$75 \div 4$

Remainder = $3$

For digit $2$, remainder $3$ means:

Unit digit = $8$

Therefore, the required unit digit is $8$

Unit Digit of Numbers Ending with $3$

Pattern of powers of $3$:

$3^1 = 3$

$3^2 = 9$

$3^3 = 27$

$3^4 = 81$

Unit digit pattern:

$3,\ 9,\ 7,\ 1$

Rule for Digit $3$

If remainder is $1$ → unit digit = $3$

If remainder is $2$ → unit digit = $9$

If remainder is $3$ → unit digit = $7$

If remainder is $0$ → unit digit = $1$

Example

Find the unit digit of $563^{26}$

Solution

The unit digit of $563$ is $3$

Now divide:

$26 \div 4$

Remainder = $2$

For digit $3$, remainder $2$ means:

Unit digit = $9$

Therefore, the required unit digit is $9$

Unit Digit of Numbers Ending with $7$

Pattern of powers of $7$:

$7^1 = 7$

$7^2 = 49$

$7^3 = 343$

$7^4 = 2401$

Unit digit pattern:

$7,\ 9,\ 3,\ 1$

Rule for Digit $7$

If remainder is $1$ → unit digit = $7$

If remainder is $2$ → unit digit = $9$

If remainder is $3$ → unit digit = $3$

If remainder is $0$ → unit digit = $1$

Example

Find the unit digit of $2587^{113}$

Solution

The unit digit of $2587$ is $7$

Now divide:

$113 \div 4$

Remainder = $1$

For digit $7$, remainder $1$ means:

Unit digit = $7$

Therefore, the required unit digit is $7$

Unit Digit of Numbers Ending with $8$

Pattern of powers of $8$:

$8^1 = 8$

$8^2 = 64$

$8^3 = 512$

$8^4 = 4096$

Unit digit pattern:

$8,\ 4,\ 2,\ 6$

Rule for Digit $8$

If remainder is $1$ → unit digit = $8$

If remainder is $2$ → unit digit = $4$

If remainder is $3$ → unit digit = $2$

If remainder is $0$ → unit digit = $6$

Example

Find the unit digit of $6368^{84}$

Solution

The unit digit of $6368$ is $8$

Now divide:

$84 \div 4$

Remainder = $0$

For digit $8$, remainder $0$ means:

Unit digit = $6$

Therefore, the required unit digit is $6$

Unit Digit of Numbers Ending with $4$ and $9$

Both digits $4$ and $9$ have a cyclicity of $2$

This means only two cases exist:

• even power

• odd power

This makes these questions much easier.

Unit Digit of Numbers Ending with $4$

Rule for Digit $4$

If power is even → unit digit = $6$

If power is odd → unit digit = $4$

Example

Find the unit digit of $564^{48}$

Solution

The unit digit of $564$ is $4$

Power $48$ is even

So, the unit digit is $6$

Therefore, the required unit digit is $6$

Unit Digit of Numbers Ending with $9$

Rule for Digit $9$

If power is even → unit digit = $1$

If power is odd → unit digit = $9$

Example

Find the unit digit of $8759^{83}$

Solution

The unit digit of $8759$ is $9$

Power $83$ is odd

So, the unit digit is $9$

Therefore, the required unit digit is $9$

These shortcut rules make unit digit questions extremely fast and scoring in competitive exams.

Practice Questions based on Unit Digit

Q.1. What is the unit digit of ( 217 ) 413 × ( 819 ) 547 × ( 414 ) 624 × ( 342 ) 812 ?

1. 2

2. 4

3. 6

4. 8

Hint: Use the concept of cyclicity to get the desired unit digit.

Solution:

Given:

(217)413 × (819)547 × (414)624 × (342)812

Solution:

The cyclicity of 7 is 4.

The cyclicity of 9 is 2.

The cyclicity of 4 is 2.

The cyclicity of 2 is 4.

According to the cyclicity theorem, divide the powers by 4 and take reminders. Otherwise, take 4.

The unit digit of (217)413 = 7413 = 7(4 × 103) + 1 = 71 = 7

The unit digit of (819)547 = 9547 = 9(2 × 273) + 1 = 91 = 9

The unit digit of (414)624 = 4624 = 4(2 × 312) = 44 = 6

The unit digit of (342)812 = 2812 = 2(4 × 203) = 24 = 6

The unit digit of the given expression will be the same as the unit digit of 7413 × 9547 × 4624 × 2812

⇒ 71 × 93 × 44 × 24

For unit digit ⇒ 7 × 9 × 6 × 6

∴ The required unit digit = 8

Hence, the correct answer is option (4).

Q.2. What is the digit in the unit's place in the number 15 ! 100 ?

1. 5

2. 7

3. 3

4. 0

Hint: Number of trailing zeroes in n ! = [ n 5 ] + [ n 25 ] + [ n 125 ] + ....... where [ ] denotes greatest integer function.

Solution:

Number of trailing zeroes in n ! = [ n 5 ] + [ n 25 ] + [ n 125 ] + ....... where [ ] denotes greatest integer function.

So, the number of trailing zeroes in 15! = [ 15 5 ] + [ 15 25 ] + [ 15 125 ] + ....... = 3 + 0 + 0 + ..... = 3

⇒ Number of zeroes in the product = 3

⇒ Unit's digit in 15 ! 100 = 0

Hence, the correct answer is option (4).

Q.3. What will be the remainder when 252 126 + 244 152 is divided by 10?

1. 4

2. 6

3. 0

4. 8

Hint: When any number is divided by 10, the unit digit will be the remainder.

Solution:

To find the remainder when 252 126 + 244 152 is divided by 10, we only need to consider the unit digit of each term in the sum.

The unit digit of 252 126 is the same as the unit digit of 2126.

Divide 126 by 4 and get remainder = 2

So, the unit digit of 252 126 is 2 2 = 4

The unit digit of 244 152 is the same as the unit digit of 4152.

If the power of 4 is even, the unit digit is 6.

So, the unit digit of 244 152 is 6.

The sum of the unit digits = 4 + 6 = 10

The remainder when 252 126 + 244 152 is divided by 10 = Remainder when 10 is divided by 10 = 0

So, the remainder is 0.

Hence, the correct answer is option (3).

Q.4. Let x = (433)24 – (377)38 + (166)54. What is the unit digit of x ?

1. 8

2. 9

3. 7

4. 6

Hint: The unit digit of the power of a number repeats itself every 4th time for the base having the last digit as 3 or 7. But the last digit for the(166)54 will always be 6 as the unit digit of a power of a number having the last digit 6 will always be 6.

Solution:

Given: x = (433)24 – (377)38 + (166)54

We can write 43324 = 433(4 × 5) + 4, 37738 = 377(4 × 9) + 2

So, the unit digit of 4334 = 1 and the unit digit of 3772 = 9

The last digit for the(166)54 will always be 6 as the unit digit of a power of a number having the last digit 6 will always be 6.

∴ x = 1 – 9 + 6 = 7 – 9 → Unit digit = 17 – 9 = 8

Hence, the correct answer is option (1).

Q.5. Find the Unit digit of 287 5 62581 .

1. 1

2. 2

3. 7

4. 4

Hint: Use the cyclicity of 7.

Solution:

The unit digit of 2587 is 7.

So, we divide the power 562581 by 4 and get the remainder of 1.

Therefore, the required unit digit is 7 1 = 7 .

Hence, the correct answer is option (3).

Q.6. The unit digit in the sum of ( 124 ) 3 72 + ( 124 ) 3 73 is:

1. 5

2. 4

3. 20

4. 0

Hint: If the power of a number ending with 4 is even, then the unit digit will be 6 and if it is odd then, the unit digit will be 4.

Solution:

Both of the numbers have a unit digit of 4 and it has a repeating cycle of 2 with unit digits 4 and 6

So in the first number power is 372 which is an even number, the unit digit of the first number will be 6.

Also in the second number power is 373 which is an odd number, the unit digit of the second number will be 4.

Therefore, the unit digit of the sum will be 6 + 4 → 0.

Hence, the correct answer is option (4).

Q.7. The digit in the unit digit in the square of 66049 is:

1. 1

2. 2

3. 3

4. 4

Hint: The square of numbers ending with 9 has the unit digit of 1.

Solution:

We know, the square of numbers ending with 9 has the unit digit of 1.

So, the unit digit in the square of 66049 is 1.

Hence, the correct answer is option (1).

Q.8. Find the unit digit in 71 × 72 × 73 × 74 × 76 × 77 × 78 × 79.

1. 1

2. 4

3. 6

4. 3

Hint: We need to multiply only the unit digits.

Solution:

The unit digit of 71 × 72 × 73 × 74 × 76 × 77 × 78 × 79

→ 2 × 2 × 2 × 2 → 6.

Hence, the correct answer is option (3).

Q.9. Find the unit digit of the sum of the first 110 natural numbers.

1. 2

2. 3

3. 5

4. 4

Hint: The sum of the first n natural numbers = n ( n + 1 ) 2 .

Solution:

The sum of the first 110 natural numbers

= 110 × ( 110 + 1 ) 2 = 110 × 111 2 = 55 × 111

So, the unit digit of the sum of the first 110 natural numbers = 5 × 1 = 5.

Hence, the correct answer is option (3).

Q.10. Find the unit digit of 432 4 12 × 499 4 31 .

1. 4

2. 3

3. 2

4. 1

Hint: Use the cyclicity of numbers to solve this.

Solution:

For the first number,

We divide 412 by 4 to get the remainder of 0.

So, the unit digit of 432 4 12 is 2 4 → 6.

Also, for the second number,

The unit digit of the number 499 is 9 and the power 431 is odd.

So, the unit digit of 499 4 31 is 9.

Therefore, the unit digit of 432 4 12 × 499 4 31 is 6 × 9 → 4.

Hence, the correct answer is option (1).

Tips and Tricks to Solve Unit Digit Questions Quickly

Unit digit questions are one of the fastest-scoring topics in quantitative aptitude because they depend on logic, cyclicity, and observation rather than lengthy calculations. In exams like SSC, Banking, NDA, CAT, and Railways, unit digit questions are frequently asked to test speed and pattern recognition.

By learning the right shortcut tricks, students can solve even large exponent problems like $7^{103}$ or $8^{245}$ within a few seconds. The most important part is understanding cyclicity and knowing which shortcut to apply.

Learn Cyclicity Patterns from $0$ to $9$

The most important trick for solving unit digit questions is memorizing the cyclicity patterns of digits from $0$ to $9$.

Every digit has a repeating pattern when raised to powers. Some digits repeat after $4$ powers, some after $2$, and some remain constant forever.

This repeating sequence is called cyclicity.

Complete Cyclicity Table for Unit Digits

Why This Table is Important

Instead of calculating the full value of large powers, we simply use the cycle.

Example

Find the unit digit of $7^{22}$

Pattern of $7$:

$7,\ 9,\ 3,\ 1$

Now divide:

$22 \div 4$

Remainder = $2$

The second number in the cycle is $9$

Therefore, the unit digit is $9$

This method saves a huge amount of time.

Shortcut Tricks for Even and Odd Powers

Digits ending in $4$ and $9$ are the easiest because they have cyclicity of only $2$.

This means we only need to check whether the power is even or odd.

Shortcut Rule for Numbers Ending with $4$

Example

Find the unit digit of $24^{18}$

The last digit is $4$

Power $18$ is even

Therefore, the unit digit is $6$

Shortcut Rule for Numbers Ending with $9$

Example

Find the unit digit of $39^{15}$

The last digit is $9$

Power $15$ is odd

Therefore, the unit digit is $9$

This shortcut is extremely useful in competitive exams.

Fast Methods for Large Exponents

Large powers like $3^{127}$ or $8^{245}$ may look difficult, but they become easy using the remainder method.

Step-by-Step Shortcut Method

Step 1: Identify the last digit of the base number

Step 2: Find the cyclicity of that digit

Step 3: Divide the exponent by the cycle length

Step 4: Use the remainder to locate the correct position in the cycle

Step 5: If remainder is $0$, use the last digit of the cycle

This is the most important method for unit digit questions.

Example

Find the unit digit of $8^{245}$

Pattern of $8$:

$8,\ 4,\ 2,\ 6$

Cycle length = $4$

Now divide:

$245 \div 4$

Remainder = $1$

The first number in the cycle is $8$

Therefore, the unit digit is $8$

Common Mistakes to Avoid in Unit Digit Questions

Many students lose marks because of small mistakes, not because the concept is difficult.

Avoiding these common errors improves accuracy immediately.

Common Mistakes Table

Important Reminder for Remainder $0$

If remainder becomes $0$, do not use the first number of the cycle.

Always use the last number of the cycle.

Example

Find the unit digit of $2^8$

$8 \div 4$

Remainder = $0$

Cycle of $2$:

$2,\ 4,\ 8,\ 6$

Use the fourth number

Therefore, the unit digit is $6$

Not $2$

This is one of the most common exam mistakes.

Special Cases in Unit Digit Problems

Some unit digit questions follow special fixed rules and do not require the full cyclicity method.

These include:

• factorial numbers

• perfect squares

• perfect cubes

• expressions involving brackets

Understanding these cases makes solving faster.

Unit Digit of Factorial Numbers

Factorial means multiplication of all natural numbers from $1$ to that number.

For example:

$5! = 5 \times 4 \times 3 \times 2 \times 1$

For any factorial greater than or equal to $5!$, the unit digit is always $0$

This happens because factorial contains both $2$ and $5$, which create $10$

And once a number ends in $0$, the unit digit remains $0$

Quick Rule Table

Example

Find the unit digit of $8!$

Since $8! \geq 5!$

Therefore, the unit digit is $0$

Unit Digit of Perfect Squares

Perfect squares can only end with certain digits.

They can have unit digits:

$0,\ 1,\ 4,\ 5,\ 6,\ 9$

They can never end with:

$2,\ 3,\ 7,\ 8$

Perfect Square Unit Digit Table

Example

Can a perfect square end with $8$?

No, because $8$ is not a possible unit digit of a perfect square.

Unit Digit of Perfect Cubes

Perfect cubes can end with any digit from $0$ to $9$

Unlike perfect squares, there is no restriction.

Example

$2^3 = 8$

$3^3 = 27$

$4^3 = 64$

Perfect cubes can have unit digits like $8,\ 7,\ 4$, etc.

This is an important exam concept.

Unit Digit of Expressions Involving Brackets

Sometimes questions come in the form:

$(23 + 17)^5$

In such cases:

First solve the bracket

Then apply the unit digit rule

Example

Find the unit digit of $(18 + 12)^3$

First solve the bracket:

$18 + 12 = 30$

Now:

$30^3$

Any number ending with $0$ always gives unit digit $0$

Therefore, the answer is $0$

Difference Between Unit Digit and Last Two Digits

Many students confuse unit digit with last two digits.

Although both are pattern-based questions, their solving methods are different.

Understanding this difference is very important.

Unit Digit vs Last Two Digits Concept

Unit digit means the digit in the one’s place only.

Last two digits means the number formed by the tens and one’s place together.

Comparison Table

These are not the same.

How to Solve Last Two Digit Questions

For last two digit questions, we often use modulus $100$

For unit digit questions, we only use cyclicity of the last digit

This makes last two digit problems slightly more advanced.

Example

Find the last two digits of $11^2$

$11^2 = 121$

Last two digits = $21$

This is different from unit digit logic.

Common Confusion Between Both Concepts

Students often apply unit digit rules directly to last two digit questions.

This gives wrong answers.

Always remember:

• unit digit → focus on one digit

• last two digits → focus on modulus $100$

This distinction is very important for competitive exams.

Applications of Unit Digit in Competitive Exams

Unit digit is one of the most frequently asked topics in number system and quantitative aptitude.

It appears in multiple chapters and helps save exam time.

Number System Aptitude Questions

Most unit digit problems belong to the number system chapter.

These include:

• powers and exponents

• cyclicity

• factorials

• perfect squares

• perfect cubes

This makes it a core topic in aptitude exams.

Simplification Problems

Many simplification questions involve multiplication and powers.

Using unit digit tricks helps solve them quickly without full calculation.

This improves speed significantly.

Algebra and Exponent-Based Questions

Expressions like:

$a^n + b^n$

or

$(x + y)^n$

often use unit digit concepts.

This makes unit digit useful beyond arithmetic.

SSC, Banking, CAT, and NDA Exam Relevance

Unit digit questions are commonly asked in:

• SSC CGL

• Banking Exams

• NDA

• Railways

• CAT

• UPSC foundation mathematics

These exams prefer logic-based fast-solving questions.

That is why unit digit is considered a very high-scoring topic.

Best Books for Unit Digit Preparation

Choosing the right book helps students improve both concept clarity and speed.

Best Books Table

These books are highly useful for mastering unit digit and number system concepts.

Important Formula and Pattern Table for Quick Revision

This quick revision table helps students revise the most important unit digit rules before exams.

Important Unit Digit Formula Table

Memorizing this table makes unit digit questions extremely fast and easy in competitive exams.

Related Quantitative Aptitude Topics

This section covers other important Quantitative Aptitude topics related to Unit Digit and Number System concepts, helping students strengthen their problem-solving speed and accuracy. It includes frequently asked topics like remainders, divisibility rules, simplification, surds and indices, and other high-scoring aptitude chapters for competitive exams.