Modular Exponentiation with Example, Finding Final digit and final two digits in given number.

 Modular Exponentiation:

Definition: Modular exponentiation is the process of repeatedly squaring and reducing a number modulo some integer, and then combining the results to find the required answer.

Modulo refers to the act of finding the remainder when a certain number is divided by another.

For examples:

5 Modulo 3 is 2. The use of this operation will be in algorithm which uses divisibility, most simply, finding whether a number is odd or even.

Exponentiation is simple raising a number to a certain power

5²=25.

Module and exponentiation are both mathematics terms, nothing related to IT field. However, your script may use this logic.

C=b^e(Mod m), where e exponent, m is modules.

To find final digit and two final digits, use modular Exponentiation method:

Universe Method:

Modular Exponentiation is a Universe method to find last digits. Use following option:

1. To find last digit use Modulo 10

2. To find last two digits use Modulo 100

3. To find last three digits use Modulo 1000 and etc

Example - 1:

What is the last digits of 78²?

Solution:

Now, number 78 deviation from 10(Universe Method-1). So we can write

= 78² Mod 10

= 6084 Mod 10

= 6084/10   here remainder is 4

= 4    

4 will be the last digit.

Example - 2

What are the last two digits of 78²?

Solution:

Now, number 78 deviation from 100 (Universe Method-2). So we can write

= 78² Mod 100

= 6084 Mod 100

= 6084/10   here remainder is 84

= 84    

84 will be the last two digits.

Example - 3

What are the last two digits of 3¹⁰⁰?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 3¹⁰⁰ with 100 using modulo method. 

So we can write.

= 3¹⁰⁰ Mod 100

= ((3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100) (3¹⁰ Mod 100)) Mod 100

First find out answer for (3¹⁰ Mod 100) = 49. Now multiply 10 times

= (49 * 49 * 49 * 49 * 49 * 49 * 49 * 49 * 49 * 49) Mod 100

= 01  

The remainder when 3¹⁰⁰ mod 100 is same as when 1/100 which is 01.      

Therefore the answer is 01.

Example - 4

What are the last two digits of 3¹⁰⁰⁰?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 3¹⁰⁰⁰ with 100 using modulo method. 

So we can write.

= 3¹⁰⁰⁰ Mod 100

= ((3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100) (3¹⁰⁰ Mod 100)) Mod 100

First find out answer for (3¹⁰ Mod 100) = 49. Now multiply 10 and divide by 100 times you will get 3¹⁰⁰ is 01.

= (01 * 01 * 01 * 01 * 01 * 01 * 01 * 01 * 01 * 01) Mod 100

= 01  

The remainder when 3¹⁰⁰⁰ mod 100 is same as when 1/100 which is 01.      

Therefore the answer is 01.

Example - 5

What are the last two digits of 7⁸¹?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 7⁸¹ with 100 using modulo method. 

So we can write.

= 7⁸¹ Mod 100

= ((7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹⁰ Mod 100) (7¹ Mod 100)) Mod 100

First find out answer for (7¹⁰ Mod 100) = 49. Now multiply 8 and divide by 100.

= (49* 49 * 49 * 49 * 49 * 49 * 49 * 49 * 7) Mod 100

= 07  

The remainder when 7⁸¹ mod 100 is same as when 7/100 which is 07.      

Therefore the answer is 07.

Example - 6

What are the last two digits of 31⁷⁶?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 31⁷⁶ with 100 using modulo method. 

So we can write.

= 31⁷⁶ Mod 100

= ((31¹⁰ Mod 100) (31¹⁰ Mod 100) (31¹⁰ Mod 100) (31¹⁰ Mod 100) (31¹⁰ Mod 100) (31¹⁰ Mod 100) (31¹⁰ Mod 100) (31⁶ Mod 100)) Mod 100

First find out answer for (31¹⁰ Mod 100) = 01 and (31¹⁰ Mod 100)=81.

= (01* 01 * 01 * 01 * 01 * 01 * 01 * 81) Mod 100

= 81  

The remainder when 31⁷⁶ mod 100 is same as when 81/100 which is 81.      

Therefore the answer is 81.

Example - 7

What are the last two digits of 135¹²³?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 135¹²³ with 100 using modulo method. 

So we can write.

= 135¹²³ Mod 100

= ((135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135¹⁰ Mod 100) (135³ Mod 100)) Mod 100

First find out answer for (135¹⁰ Mod 100) = 25 and (135³ Mod 100)=75.

= (25 * 25 * 25 * 25 * 25 * 25 * 25 * 25 * 25 * 25 * 25 * 25 * 75) Mod 100

= 75  

The remainder when 135¹²³ mod 100 is same as when 75/100 which is 75.      

Therefore the answer is 75.

Example - 8

What are the last two digits of 3²⁰¹⁹?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

When ever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 3²⁰¹⁹ with 100 using modulo method. 

So we can write.

= 3²⁰¹⁹ Mod 100

= ((3⁵⁰⁰ Mod 100) (3⁵⁰⁰ Mod 100) (3⁵⁰⁰ Mod 100) (3⁵⁰⁰ Mod 100) (3¹⁹ Mod 100)) Mod 100

First find out answer for (3⁵⁰⁰ Mod 100) = 01 and (3¹⁹ Mod 100)=67.

= (01 * 67) Mod 100

= 67  

The remainder when 3²⁰¹⁹ mod 100 is same as when 67/100 which is 67.      

Therefore the answer is 67.

Example - 9

What are the last two digits of 2¹⁹⁹⁷?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

Whenever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 2¹⁹⁹⁷ with 100 using modulo method. 

So we can write.

= 2¹⁹⁹⁷ Mod 100

= ((2⁵⁰⁰ Mod 100) (2⁵⁰⁰ Mod 100) (2⁵⁰⁰ Mod 100) (2⁴⁰⁰ Mod 100) (2⁹⁷ Mod 100)) Mod 100

First find out answer for (2⁵⁰⁰ Mod 100) = 76,  (2⁴⁰⁰ Mod 100)=76 and 

(2⁹⁷ Mod 100)=72.

= (76 * 76 * 72) Mod 100

= 72 

The remainder when 2¹⁹⁹⁷ mod 100 is same as when 72/100 which is 72.      

Therefore the answer is 72.

Example - 10

What are the last two digits of 168¹⁰⁵⁷?

Solution:

The basic things required to answer this kind of question is that you must have a good knowledge of Modulo theorem, addition, multiplication, division and subtraction. 

Whenever a number is divided by 10 we will get the units digits, if divided by 100 we will get the last two digits i.e., units and tenth. If divided by 1000 we will get the last 3 digits of the number i.e., units, tenth and hundredth digit of the number and so on.

Now how to divide 168¹⁰⁵⁷ with 100 using modulo method. 

So we can write.

= 168¹⁰⁵⁷ Mod 100

= ((168⁵⁰⁰ Mod 100) (168⁵⁰⁰ Mod 100) (168⁵⁷ Mod 100)) Mod 100

First find out answer for (168⁵⁰⁰ Mod 100) = 76,  (168⁵⁷ Mod 100)=68.

= (76 * 76 * 68) Mod 100

= 68 

The remainder when 168¹⁰⁵⁷ mod 100 is same as when 68/100 which is 68.      

Therefore the answer is 68.

If you need more examples, kindly drop a messages in comment section. 

Thank you.