Question

Suppose a and b are integers, a ≡ 11 (mod 19) and b ≡ 3 (mod 19). Find an integer c with 0≤c≤18 such that
(a)c≡13a(mod 19).
(b)c≡8b(mod 19).
(c)c≡a-b(mod 19).
(d)c≡7a+ 3b(mod 19).
(e)c≡2a^2+ 3b^2(mod 19).
(f)c≡a^3+ 4b^3(mod 19).

Answer 1

a.) $13*11=143$

$143=10(mod 19)$

b.)$8*3=24$

$24=5(mod 19)$

c.)$a-b=11-3=8(mod19)$

d.)$77+9=1+9=10(mod19)$

e.)$2*11^2+3*3^2= 2*7+3*9=14+27=41=3(mod 19)$

f.) $11^3+4*3^3=7*11+3*3*9=1+5=6(mod19)$