Program to find the last two digits of x^y

The task is to find the last two digits of x^y.

Since the digits with which it can end are 0-9, Hence this problem can be divided into 5 cases:

1. Case 1: when x ends with 1

   For finding the last two digit of a number, when the number ends with 1 then we have to do following steps shown as in the figure.

   Example: 21^48

   

   So, Last two digit of 21^48 is 81.

   Example: 31^35
   

   So, Last two digit of 31^35 is 51.

2. Case 2: when x ends with 3, 7, 9

   For finding the last two digit of a number, when the number ends with 3, 7, 9 then we have to apply cyclicity concept to convert the last digit as a 1.

   cyclicity of 3:

   3^1 = 3
   3^2 = 9
   3^3 = 7
   3^4 = 1

   cyclicity of 7:

   7^1 = 7
   7^2 = 9
   7^3 = 3
   7^4 = 1

   cyclicity of 9:

   9^1 = 9
   9^2 = 1

   Example1: 23^34
   Solution:

   • Last digit of 23^34 is 3 so, we use cyclicity of 3 .
   • 3^4 gives 1 so, we take 23^4
   • ((23)^4)^8 * (23)^2
   • last two digit of (23)^4) is 41, so we take (41)^8 and solve according to the given diagram.

   

   • So last digit of (41)^8 is 21 .
   • solve (23)^2, the last digit of (23)^2 is 29.
   • Now multiply last digit of (41)^8 i.e 21 with the last digit of (23)^2 i.e 29
   • i.e 21 * 29 = 609
   • So, Last two non zero digit of 23^34 is 09.

   Example2: 37^45
   Solution:

   • Last digit of 37^45 is 7 so, we use cyclicity of 7 .
   • 7^4 gives 1 so, we take 37^4
   • ((37)^4)^11 * (37)^1
   • last two digit of (37)^4) is 61, so we take (61)^11 and solve according to the diagram.

   

   • So last digit of (61)^11 is 61 .
   • solve (37)^1, the last digit of (37)^1 is 37.
   • Now multiply last digit of (61)^11 i.e 61 with the last digit of (37)^1 i.e 37
   • i.e 61 * 37 = 2257
   • So, Last two non zero digit of 37^45 is 57.

   Example3: 59^22
   Solution:

   • Last digit of 59^22 is 9 so, we use cyclicity of 9 .
   • 9^2 gives 1 so, we take 59^2
   • ((59)^2)^11
   • last two digit of (59)^2 is 81, so we take (81)^11 and solve according to the diagram.

   

   • So last digit of (81)^11 is 81 .
   • So, Last two non zero digit of 59^22 is 81.
3. Case 3: when x ends with 2, 4, 6, 8

   For finding the last digit of a number ends with 2, 4, 6, 8; We use number 76 which is a type of magic number because its square, cube and etc contain last 2 digit numbers as itself i.e 76.

   Take an example :

   square of 76 = 5776, its last two digit =76
   cube of 76 = 438976, its last two digit=76

   So we take two cases:

   1. if (2^10)^even power then it always return 76 .
   2. if (2^10)^odd power then it always return 24 .

   Steps for finding last two digits

   • Firstly, convert given number in to these formats if (2^10)^power.
     Here power will be odd or even according to the question.
   • Now, check power will be odd or even.
   • if power is odd then its value will be 24.
   • if power is even then its value will be 76.

   Examples

   Example1: Find last 2 digit of 2^453.

   Solution:

   • Step 1:- conversion
     2^453 = (2^10)^45 * 2^3
   • Step 2:- odd power so we take 24
     = 24 * 8
     = 192

   So, Last two non zero digits of 2^453 are 92.

   Example2: Find last 2 digits of 4^972.

   Solution:

   • step 1:- conversion
     4^972 = (2^2)^972
     = 2^1944
     = (2^10)^194 * 2^4
   • step 2:- even power so, we take 76
     = 76 * 16
     = 1216

   So, Last two non zero digits of 4^972 are 16.

   Example3: Find last 2 digits of 6^600.

   Solution:

   • step 1:- conversion
     6^600 = (2)^600 * (3)^600
     = (2^10)^60 * ((3)^4)^150 {Apply case 2 in (3)^600}
   • step 2:- (2^10)^60 has even power so, we take 76 as the last digit
   • step 3:- Solve ((3)^4)^150, we get 01 as the last digit
   • step 4:- last digit of (2^10)^60 i.e 76 multiply with the last digit of ((3)^4)^150 i.e 01
   • step 5:- i.e 76 * 01 = 76

   So, Last two non zero of 6^600 is 76.

   Example4: Find last 2 digits of 8^330.

   Solution:

   • step 1:- conversion
     8^33 = (2^3)^110
     = (2)^330
   • step 2:- (2^10)^33 has odd power so, we take 24 as the last digit

   So, Last two non zero digits of 8^330 are 24.

4. Case 4: when x ends with 5

   For finding the last two digit of a number, when the number ends with 5 then we have to follow the table which is given below.

   

   Example1: Find last 2 digit of 25^25.

   Solution:

   • first digit of number is 2 i.e even
   • Last digit of a power is 5 i.e odd
   • Now, even-odd combination gives last digit as a 25

   So, the last two non zero digits of 25^25 are 25.

   Example2: Find last 2 digit of 25^222.

   Solution:

   • first digit of number is 2 i.e even
   • Last digit of a power is 2 i.e even
   • Now, even-even combination gives last digit as a 25

   So, the last two non zero digits of 25^222 are 25.

   Example3: Find last 2 digit of 165^222.

   Solution:

   • first digit of number is 1 i.e odd
   • Last digit of a power is 2 i.e even
   • Now, odd-even combination gives last digit as a 25

   So, the last two non zero digits of 165^222 are 25.

   Example4: Find last 2 digit of 165^221.

   Solution:

   • first digit of number is 1 i.e odd
   • Last digit of a power is 1 i.e odd
   • Now, odd-odd combination gives last digit as a 75

   So, the last two non zero digits of 165^221 are 75.

5. Case 5: when x ends with 0

   For finding the last two digit of a number, when the number ends with 0 then we have to check next digit and according to the digit calculate the last digit.

   Example: Find last 2 digit of 150^221.

   Solution:

   • 150 last digit is 0 so we check next digit i.e 5 and apply case 4
   • first digit of number is 1 i.e odd
   • Last digit of a power is 1 i.e odd
   • Now, odd-odd combination gives last digit as a 75

   So, the last two non zero digits of 165^221 is 75.