+249 1. Let n be a positive integer and a be an integer such that a is its own inverse modulo n . What is the remainder when a^2 is divided by n?
2. Find the smallest positive integer that satisfies the system of congruences \(\begin{align*} N &\equiv 2 \pmod{11}, \\ N &\equiv 3 \pmod{17}. \end{align*}\)
3. Find the smallest positive N such that \(\begin{align*} N &\equiv 6 \pmod{12}, \\ N &\equiv 6 \pmod{18}, \\ N &\equiv 6 \pmod{24}, \\ N &\equiv 6 \pmod{30}, \\ N &\equiv 6 \pmod{60}. \end{align*}\)
4. How many positive integers less than or equal to 6*7*8*9 solve the system of congruences \(\begin{align*} m &\equiv 5 \pmod{6}, \\ m &\equiv 4 \pmod{7}, \\ m &\equiv 3 \pmod{8}, \\ m &\equiv 3 \pmod{9}. \end{align*}\)
5. Find the smallest positive N such that \(\begin{align*} N &\equiv 3 \pmod{4}, \\ N &\equiv 2 \pmod{5}, \\ N &\equiv 6 \pmod{7}. \end{align*}\)
Thank you for all of your help in advance! Sorry for so many problems!
