1. Find all numbers $r$ for which the system of congruences
\begin{align*}
x &\equiv r \pmod{6}, \\
x &\equiv 9 \pmod{20}, \\
x &\equiv 4 \pmod{45}
\end{align*}has a solution.
2. Let $x$ and $y$ be integers. Show that $9x + 5y$ is divisible by $19$ if and only if $x + 9y$ is divisible by $19.$
3.
a) Show that $n(2n + 1)(7n + 1)$ is divisible by 6 for all integers $n$.
b) Find all integers $n$ such that $n(2n + 1)(7n + 1)$ is divisible by 12.
4.
a) Show that the sum of 11 consecutive integers is always divisible by 11.
b) Show that the sum of 12 consecutive integers is never divisible by 12.
5. The units digit of a perfect square is 6. What are the possible values of the tens digit?
Four positive integers $p,q,r,s$ satisfy the following equations:
\begin{align*}
pq+2p+q&=348 \\
qr+4q+3r&=373 \\
rs+8r+6s&=544
\end{align*}
What are $p,q,r,$ and $s$?
TIA!
1. The set of all n that works is n congruent to 18 mod 30.
5. The odd digits 1, 3, 5, 7, 9 are the possible values of the tens digit.
