## Discrete Mathematics and Its Applications, Seventh Edition

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.