# May 6th Review Problems

1. Solve for $$x$$: $$10^{35} x \equiv 4 \mod{7}$$
2. Find the smallest non-negative $$x$$ such that: $$x \equiv 13 \mod{25} \\ x \equiv 7 \mod{9}$$
3. To the previous problem add the condition: $$x \equiv 1 \mod{4}$$
4. Show that any list of 20 integers must have two whose difference is divisible by 13.
5. Seventy cars sit on a parking lot. Thirty have stereo systems, 30 have A/C, 40 have sun roofs. Thirty of these cars have at least 2 of these 3 options. Ten cars have all three. How many have at least one option? How many have only one option?
6. What is the coefficient of $$x^{25}$$ in the binomial expansion of $$(2x - \frac{3}{x^2})^{58}$$?
7. In how many ways can two white rooks, two black bishops, eight black pawns, and eight white pawns be placed on a prescribed 20 squares of a chessboard?
8. Prove that $$\binom{2n}{2} = 2\binom{n}{2} + n^2$$ using logic not formula.

Is it anyone's birthday?