How to find Modular Arithmetic ?

Question

(a) Compute (7*11*711*777) mod 13 by modular arithmetic.
(b) Compute (666^77) mod 11.

Please tell me the procedure to solve such a question not just Answer  .I know the answer.

Answer 1

(i) 7*11*711*777%13

try to minimize individual number

=(7*11*9*10)%13  (here 711mod13=9 and 777mod 13=10)

=(77*90)%13

=(-1 * -1)%13

=1 Ans

do other in same try to minimize

Answer 2

1. is done simply using reminder theorem application based

(77*711*777) mod 13 which means it can be write like as ((77mod13)*(711mod13)*(777mod13))mod13

by which u get (12*9*10)mod13 which gives u remider 1 ANS

and for 

(ii) (666)^77 

u do like( (666mod11)* (666mod11)*(666mod11)-----------------------77times) then u got

(6*6*6-------------------------------77times) which means 6^77 then for that ((6^2 mod 11) *(6^2 mod 11) *(6^2 mod11) *6 mod 11)^11 then u got  8^11 mod 11 then write like as ((8^3 mod 11)*(8^3mod11)*(8^3mod11)*(8^2 mod 11)) then u got finally   ANS is 8

Comments

m not getting the 2nd part ,please clarify!