Abstract Algebra: Theory and Applications · Problem 13

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · chinese-remainder-theorem-rings

Problem

Use the Chinese Remainder Theorem to solve each of the following systems of congruences. (a)

\begin{aligned} x &\equiv 2 \pmod{5}\\ x &\equiv 6 \pmod{11} \end{aligned}

(b)

\begin{aligned} x &\equiv 3 \pmod{7}\\ x &\equiv 0 \pmod{8}\\ x &\equiv 5 \pmod{15} \end{aligned}

(c)

\begin{aligned} x &\equiv 2 \pmod{4}\\ x &\equiv 4 \pmod{7}\\ x &\equiv 7 \pmod{9}\\ x &\equiv 5 \pmod{11} \end{aligned}

(d)

\begin{aligned} x &\equiv 3 \pmod{5}\\ x &\equiv 0 \pmod{8}\\ x &\equiv 1 \pmod{11}\\ x &\equiv 5 \pmod{13} \end{aligned}

Hint. (a)

x \equiv 17 \pmod{55}

; (c)

x \equiv 214 \pmod{2772}