Unit digit questions-cyclicity of numbers

How to find the unit digit of a number raised to a power

Learn to find the unit digit of a number raised to some power using the concepts like cyclicity of numbers in simple way. I will also tell you some important math tricks and shortcuts so that you can solve the unit digit questions in seconds.

If I say to you to find the unit digit of 646^243

It seems very difficult to find out the answer to the above question?

Probably yes now, but I promise you that after reading following you will solve this with in 5 seconds.

Please read the complete article, I promise you will solve all type of questions after reading this article completely. The article is written to give in-depth knowledge along with shortcut tricks to calculate even faster.

Before answering the above question, lets first we understand such types of the problem thoroughly then we will come back at the above question.

If a number is written in the form of  X^Y, here  X is base whose unit digit may vary from 0 to 9 and Y is exponent whose unit digit also may vary from 0 to 9. 

CASE 1: If the base has a unit digit equal to 1.

Look at the pattern of the unit digit. Only one digit is coming in unit place and that is 1.

1^1 = 1

1^2 = 1

1^3 = 1

1^4 = 1

It means if the base has 1 in its unit place on raising its power to any number there will be always 1 in unit place of the answer, no matter what the exponent is.

Here only one number( that is 1) is appearing at the unit place so here cyclicity is 1.

Example: find the unit digit of 3161^24

Here the base is 3161 and its unit digit is 1

Hence unit digit in the answer will be 1.

CASE 2: If the base has a unit digit equal to 2.

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32       

2^6 = 64

2^7 = 128

2^8 = 256

Look at the pattern of the unit digit. Only four numbers are coming in unit place and those are 2, 4, 8, and 6. And notice they are following sequence 2, 4, 8, 6 and after 6, 2 is coming again and the same cycle is repeated.

• Here four numbers (2, 4, 8, and 6) are appearing at the unit place so cyclicity is 4.
•  It means 2^1, 2^5, 2^9 , 2^13 and so on. All have unit digit equal to 2. In general, all the exponent numbers which are in the form of 2^(4n+1) have unit digit equal to 2. In other words, we can say that if on dividing the exponent by 4  if we get a remainder equal to 1 it means there will be 2 in unit place in the answer.

 ( Note: we are dividing by 4 because cyclicity is 4)

• Similarly, 2^2, 2^6, 2^10, 2^14 and so on. all have unit digit equal to 4. In general, all the exponent numbers which are in the form of 2^(4n+2) have unit digit equal to 4. In other words, we can say that if on dividing the exponent by 4  if we get a remainder equal to 2 it means there will be 4 in unit place in the answer.
•  Similarly, 2^3, 2^7, 2^11, 2^15 and so on.  all have unit digit equal to 8. In general, all the exponent numbers which are in the form of 2^(4n+3) have unit digit equal to 8. In other words, we can say that if on dividing the exponent by 4 if we get a remainder equal to 3 it means there will be 8 in unit place in the answer.
• Similarly, 2^4, 2^8, 2^12, 2^16 and so on. all have unit digit equal to 4. In general, all the exponent numbers which are in the form of 2^(4n) have unit digit equal to 6. In other words, we can say that if on dividing the exponent by 4  if we get a remainder equal to 0 it means there will be 6 in unit place in the answer.

NOTE: 'n' can take only integer value (i.e. 1,2,3…………)

Still, if you didn’t get don’t worry, the example below will solve your all the queries.

Now we will use the above observation to answer the questions

Example: Find the unit digit of 152^343

Here the base is 152 and the unit digit of Base is 2. So only four numbers are possible at the unit place and those are 2, 4, 8 and 6.

Now, focus on the only exponent and that is 343.

On Dividing 343 by 4, we get the remainder equal to 3. So, there will be 8 at the unit place in the answer of 152^343.

CASE 3: if the base has unit digit equal to 3.

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

3^7 = 2187

Look at the pattern of the unit digit. Only four numbers are coming in unit place and those are 3, 9, 7,

and 1 also Notice they are following sequence 3, 9, 7, 1 and after 1, 3 is coming again and the same cycle is repeated.

Here four numbers (3, 9, 7, and 1) may appear at the unit place so cyclicity is 4.

• On dividing the exponent by 4 if we get a remainder equal to 1 it means there will be 3 in unit place in the answer.On dividing the exponent by 4 if we get a remainder equal to 2 it means there will be 9 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 3 it means there will be 7 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 0 it means there will be 1 in unit place in the answer.

Example: find unit digit of 323^133

Here the base is 323 and its unit digit is 3 and the exponent is 133

On dividing the 133 by 4 we get a remainder equal to 1.

Hence unit digit in the answer will be 3.

CASE 4: If the base has a unit digit equal to 4.

Look at the pattern of the unit digit. Only two numbers are coming in unit place and these are 4 and 6.

4^1 = 4

4^2 = 16

4^3 = 64

4^4 = 256

Here two numbers (4 and 6) may appear at unit place so cyclicity is 2.

On dividing the exponent by 2 if we get a remainder equal to 1 it means there will be 4 in unit place in the answer.

On dividing the exponent by 4 if we get a remainder equal to 0 it means there will be 6 in unit place in the answer.

NOTE: We divide by 2 because cyclicity is 2.

Example: find the unit digit of 414^34

Here the base is 414 and it is unit digit is 3 and exponent is 34.

On dividing 34 by 2 we get a remainder equal to 0.

Hence unit digit in the answer will be 6.

CASE 5: If the base has a unit digit equal to 5.

Look at the pattern of the unit digit. Only two numbers are coming in unit place and these are 4 and 6.

5^1 = 5

5^2 = 25

5^3 = 125

5^4 = 625

Here only one number (That is 5) may come at unit place so cyclicity is 1.

It means if the base has 5 in its unit place on raising its power to any number there will be always 5 in unit place of the answer, no matter what the exponent is.

Example: find the unit digit of 4155^104

Here the base is 4155 and its unit digit is 5

Hence unit digit in the answer will be 5.

CASE 6: If the base has a unit digit equal to 6.

Look at the pattern of the unit digit. Only two numbers are coming in unit place and these are 4 and 6.

6^1 = 6

6^2 = 36

6^3  = 216

6^4 = 1296

Here only one number (That is 6) may come at unit place so cyclicity is 1.

It means if the base has 6 in its unit place on raising its power to any number there will be 6 in unit place of the answer, no matter what the exponent is.

Example: find the unit digit of 316^145

Here the base is 316 and its unit digit is 6

Hence unit digit in the answer will be 6.

CASE 7: If the base has unit digit equal to 7.

7^1 = 7

7^2 = 49

7^3 = 343

7^4 = 2401

7^5 = 16807

7^6 = 117649

7^7 = 823543

7^8 = 5764801

Look at the pattern of the unit digit. Only four numbers are coming in unit place and those are 7, 9, 3, and 1. And notice they are following sequence 7, 9, 3, 1 and after 1, 7 is coming again and the same cycle is repeated.

Here four numbers (3, 1, 7, and 9) may come at the unit place so cyclicity is 4.

• On dividing the exponent by 4 if we get a remainder equal to 1 it means there will be 7 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 2 it means there will be 9 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 3 it means there will be 3 in unit place in the answer.
•  On dividing the exponent by 4 if we get a remainder equal to 0 it means there will be 1 in unit place in the answer.

Example: find unit digit of 3237^133

Here the base is 323 and its unit digit is 3 and the exponent is 133

On dividing 133 by 4 we get a remainder equal to 1.

Hence unit digit in the answer will be 7.

CASE 8: if the base has unit digit equal to 8.

8^1 = 8

8^2 = 64

8^3 = 512

8^4 = 4096

8^5 = 32768

8^6 = 262144

8^7 = 2097152

8^8 = 16777216

Look at the pattern of the unit digit. Only four numbers are coming in unit place and those are 8, 4, 2, and 6. And notice they are following sequence 8, 4, 2, 6 and after 6, 2 is coming again and the same cycle is repeated.

Here four numbers (8, 4, 2, and 6) may come at the unit place so cyclicity is 4.

• On dividing the exponent by 4 if we get a remainder equal to 1 it means there will be 8 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 2 it means there will be 4 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 3 it means there will be 2 in unit place in the answer.
• On dividing the exponent by 4 if we get a remainder equal to 0 it means there will be 6 in unit place in the answer.

Example: find unit digit of 36838^232

Here the base is 36838 and its unit digit is 8 and the exponent is 232

On dividing 232 by 4 we get a remainder equal to 0.

Hence unit digit in the answer will be 6.

CASE 9: If the base has a unit digit equal to 9.

Look at the pattern of the unit digit. Only two numbers are coming in unit place and these are 4 and 6.

9^1 = 9

9^2 = 81

9^3 = 729

9^4 = 6561

Here two numbers (9 and 1) are coming at unit place so cyclicity is 2.

• On dividing the exponent by 2 if we get a remainder equal to 1 it means there will be 9 in unit place in the answer.
• On dividing the exponent by 2 if we get a remainder equal to 0 it means there will be 1 in unit place in the answer.

Example: find the unit digit of 889^889

Here the base is 889 and its unit digit is 9 and exponent is 889.

On dividing 889 by 2 we get a remainder equal to 1.

Hence unit digit in the answer will be 9.

CASE 10: If the base has a unit digit equal to 0.

Look at the pattern of the unit digit. Only two numbers are coming in unit place and these are 4 and 6.

0^1 = 0

0^4 = 0

0^3 = 0

0^4 = 0

Cyclicity is 1 because only one number appear at unit place.

It means if the base has 0 in its unit place on raising its power to any number there will be always 0 in unit place of the answer, no matter what the exponent is.

Example: find the unit digit of 3120²⁴

Here the base is 3120 and its unit digit is 0

Hence unit digit in the answer will be 0.

Summary

Unit digit and cyclicity of numbers

Now coming back to the question which is written in the starting of the blog.

find the unit digit of 646²⁴³

Tags: Unit digit, Unit digit questions


I hope now I don’t need to tell the answer to this question. You already know it.

Tell me your answer below in the comment section.