Exponential Test

Not a tough problem but worth sharing.. Here is the solution $243^{69}$ can be written as $3^{345}$ Now as we see the patern of exponential form of 3, the unit place has a pattern of 3,9,7,1 Therefore there are only four possible cases. So we divide 345 by 4. If remainder = 1 then Answer is 3 If remainder = 2 then Answer is 9 If remainder = 3 then Answer is 7 If remainder = 0 then Answer is 1 In this case as the remainder is 1... Hence the answer is 3 ;)

243 to the power that is a multiple of 10 has the unit digit 9. Divide that by 3 you will get 3.

unit place start from 7^1 till 7^5 make a pattern (7,9,3,1,7).

7^(3+69)=7^72 72/5 mod 2

So we use the 2nd from 7 that is 3.

we want 3^(345) mod 10. or, 3(9^172) mod 10. which is 3(1) mod 10. hence, 3.