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Hello friends welcome to lecture in this lecture I will give some basic knowledge about euler’s theorem so europe proposed two theorems first water and second version the first wooden say that when air is to Phi n which is congruent to 1 mod n second version is H 2 K multiplied

By Phi n plus 1 which is congruent to a mod n so these are basically two roots which will help us to solve the example so let’s go to the first example which is 6 this 224 mod 35 so as you can see here the Phi function I have taught you

In the lecture number 5 of my so here n is 35 so what will be 535 will be equal to 75 7 minus 1 into 5 minus 1 will be equal to 16 to 4 which is equal to 24 so Phi 35 is equal to 24

So now just replace 24 with 535 which will be equal to 6 days to 535 mod 35 so as you can relate this with first version so here it is 2 Phi n which is congruent to 1 mod n so what will be the

Answer is 6 6 2 5 35 more 35 will be equal to always 1 it because if you see this 6 is 2 5 35 135 and the first version if they both are similar then answer will be always equal to 1 and in second version the air is 2 K multiplied

By Phi n plus 1 which is congruent to a model here answer will be always equal to a so now let us solve example 2 so now 2 is 20 raised to 62 mod 77 so first we have to find 577 which will be equal to 7 into 11 7 minus

1 into 11 minus 1 which we build 6 into 10 will be equal to 60 so as you can see the number 62 is greater than 60 so now just see the second version so here we have to form the such a way that 62 is formed with the help of second version

So oil tea I will just teach you how to relate with it now here a is 20 and n is 77 so 20 into 2 raise to K multiplied by 60 plus 1 mod 77 so here you are to take K value as 1 because if we have to form

62 if we will take K value s 2 it will be more than 120 so K value will be equal to 1 now the equation formed is 20 raised to 61 mod 77 but we have to form 20 raised to 62 so we have to just

Multiply the 20 more 77 so 20 raised to 60 one more 77 as you can see the second version which will be equal to a so here is 20 so answer will be 20 into 20 mod 77 is equal to 20 so 20 x 20 more 77 so

Will be 400 mod 77 which will be equal to 15 thank you

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