Question

Exercise 2 (5 points Find the smallest positive solution and the general solution to the system ΧΞ2 (mod 5) and r Ξ 3 (mod 7). 1 (mod 3),

Find the smallest positive solution and the general solution to the system x ≡ 1 (mod 3), x ≡ 2 (mod 5) and x ≡ 3 (mod 7).

Answer 1

X = 1(mod3) x = 2(mod5) x = 3(mod7) A_r m_1 = 3 m_2 = 5 m_3 = 7 they are relatively prime to each other we are chaines remainder theorem a_1 = 1 a_2 = 2 a_3 = 3 m = m1 times m_2 times m_3 = 3 times 5 times 7 = 105 That is m_1 = m/m_1 = 35 m_2 = m/m_2 = 21 m_3 = m/m_3 = 15 we find m_1^- 1 m_2^- 1 m_3^- 1 w.r.t mod m_1, m_2, m_3 (1) m_1. m_1^- 1 = 1 mod m1 35. m_1^- 1 = 1 (mod 3) 2 m_1^- 1 = 1(mod 3) m_1^- 1 = 2(mod 3) m_1^- 1 = 2 (2) m_2 m_2^1 (mod 2) 21 m_2^- 1 = 1 (mod 5) m_2^- 1 = 1