Answer
a) 10
b) 8
c) 0
d) 9
e) 6
f) 2
Work Step by Step
Find the integer $c$ with 0≤$c$≤12 s.t.
a)
$c≡9(4$ mod 13) mod 13=36 mod 13
$c≡2(13)+10$ mod 13, so $c$ is within the desired domain is 10.
b)
$c≡11(9$ mod 13) mod 13=99 mod 13
$c≡7(13)+8$ mod 13, so $c$ is within the desired domain is 8.
c)
$c≡(4$ mod 13+9 mod 13) mod 13=(13 mod 13) mod 13=13 mod 13
$c≡0(13)+0$ mod 13, so $c$ within the desired domain is 0.
d)
$c≡(2(4$ mod 13)+3(9 mod 13)) mod 13=(8+27) mod 13=35 mod 13
$c≡2(13)+9$ mod 13, so $c$ is within the desired domain is 9.
e)
$c≡((4$ mod 13)$^2$+(9 mod 13)$^2$) mod 13
=(16 mod 13+81 mod 13) mod 13
=64 mod 13−729 mod 13=97 mod 13
$c≡7(13)+6$ mod 13, so c is within the desired domain is 6.
f)
$c≡((4$ mod 13)$^3$−(9 mod 13$^3$)) mod 13=64 mod 13−729 mod 13
=665 mod 13
$c≡$51(13)+2 mod 13, so c is within the desired domain is 2.
