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?