Find the remainder when

Question

(13)⁷³+(14)³ is divided by 11

Answer 1

This is a modular arithmetic problem. Note that (ab)mod 11  - a mod11 * b mod 11. Lets do 14³ first

this allows you to break down 13^73 into 13^36 * 13^36 * 2. Do the mod 11 of each term individually, multiply the answers and then take mod11 again of the final result. It is easy to make a mistake and you can check it by simply computing the modular arithmetic computations in your browser search window. For example type (13^9)mod11 = and you get the answer. I don't guarantee my arithmetic here so work through it yourself.

 

14mod11 = 3 so (14³) mod 11 = (3³)mod11 = 27mod11 = 5

 

13mod11 = 2 so (13)^73 mod 11 = 2^73 mod11 = 2^36 mod11 * 2^36 mod11 * 2

2^36 = (2^9 * 2^9 *2^9* 2^9) mod 11 recall 2^9 = 512 so 2^9 mod 11 equal 512 mod 11 = 6

so 2^36 mod 11 = 6*6*6*6 mod11 = 9

 

So then 2^73 mod 11 = (9*9*2) mod 11 = 8

 

The final answer seems to be (8+5)mod11 = 2 It is easy to make a mistake in this calculation and there could be a shortcut way to do it that I did not see off hand.