Question

Suppose that a and b are integers, a ≡ 11 (mod 19), and b ≡ 3 (mod 19). Find the integer c with 0 ≤ c ≤ 18 such that c ≡ a3 + 4b3 (mod 19).

Answer 1

is that supposed to be
\[c =a^3 + 4b^3 (\mod19)\]

Answer 2

ya i forgot

Answer 3

if a = 11 mod 19,
then a^3 = 11^3 mod 19 = 1331 mod 19 = 1 mod 19

Answer 4

because 1331 = 19*70 + 1

Answer 5

remainder of 1 upon dividing 11^3 by 19

Answer 6

ok but what about the +4b^3(mod19) part

Answer 7

if b= 3 mod 19, then b^3 = 3^3 mod 19 = 27 mod 19 = 8 mod 19

then
4b^3 = 4* 8 mod 19 = 32 mod 19 = 13 mod 19

Answer 8

now you can add those two simplifed expressions

Answer 9

so ok, this is what i should add?
(1 mod 19)+(13 mod 19) right

Answer 10

yes

Answer 11

wouldn't the answer be 14 mod 19?

Answer 12

or you can just say the integer is 14

Answer 13

thanks alot @perl

Answer 14

your welcome :)