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Suppose that a and b are integers, a ≡ 11 ( mod 19), and b ≡ 3 ( mod 19 ) .find integer c with0 ≤ c ≤ 18 such that

a.c ≡ 13 a ( mod19).

b.c ≡ 8 b( mod19).

c.c ≡ a − b( mod19).

d.c ≡ 7 a + 3 b( mod19).

e.c ≡ 2 a 2 + 3 b 2 ( mod19). f) c ≡ a 3 + 4 b 3 ( mod19).

Respuesta :

LammettHash

a. Since [tex]a\equiv11\pmod{19}[/tex], we have

[tex]c\equiv13a\equiv-6a\equiv-66\equiv10\pmod{19}[/tex]

so [tex]c=10[/tex]

b. [tex]b\equiv3\pmod{19}[/tex], so [tex]8b\equiv24\equiv5\pmod{19}[/tex], so [tex]c=5[/tex].

c. [tex]a-b\equiv11-3\equiv8\pmod{19}[/tex], so [tex]c=8[/tex].

d. [tex]7a+3b\equiv86\equiv10\pmod{19}[/tex]

e. [tex]2a^2+3b^2\equiv269\equiv3\pmod{19}[/tex]

f. [tex]a^3+4b^3\equiv1439\equiv14\pmod{19}[/tex]

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