Find $x$ where $x$ is the least non-negative residue modulo n of the following:
(a) $2789 + 2788 ≡ x \pmod{2787}$
(b) $2789 × 2788 ≡ x \pmod{2787}$
(c) $5201 + 5211 ≡ x \pmod{5200}$
(d) $5201 × 5211 ≡ x \pmod{5200}$
(e) $5198 + 5188 ≡ x \pmod{5200}$
(f) $5198 × 5180 ≡ x \pmod{5200}$
For these exercises we use Proposition (3.6)
If $a ≡ b \pmod{n}$ and $c ≡ d \pmod{n}$ then
(i) $a + c ≡ (b + d) \pmod{n}$
(ii) $ac ≡ bd \pmod{n}$
(a) $2789 + 2788 \equiv 2 \pmod{2787} + 1 \pmod{2787} \equiv 3 \pmod{2787}$
So $x=3$.
(b) $2789 \times 2788 \equiv 2 \pmod{2787} \times 1 \pmod{2787} \equiv 2 \pmod{2787}$
So $x=2$.
(c) $ 5201 + 5211 \equiv 1 \pmod{5200} + 11 \pmod{5200} \equiv 12 \pmod{5200}$
So $x=12$.
(d) $ 5201 \times 5211 \equiv 1 \pmod{5200} \times 11 \pmod{5200} \equiv 11 \pmod{5200}$
So $x=11$.
(e) $5198 + 5188 \equiv (-2) \pmod{5200} + (-12) \pmod{5200} \equiv (5200-24) \pmod{5200} \equiv 5186 \pmod{5200}$
So $x=5186$.
(f) $5198 \times 5180 \equiv (-2) \pmod{5200} \times (-20) \pmod{5200} \equiv 40 \pmod{5200}$
So $x=40$.