Review Exercises (Set A)

  1. Prove that the product of two numbers that take the form $5k+3$, where $k$ is some integer, takes the form $5k+4$.  

  2. Prove the irrationality of the following values:

    1. $\sqrt{5}$
    2. $\sqrt[3]{3}$
    3. $\log_7 3$
    4. $\sqrt{3} + \sqrt{7}$, assuming $\sqrt{3}$ is known to be irrational
  3. Use induction to prove the following is true for every positive integer $n$:

    $$1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \cdots + n \cdot n! = (n+1)! - 1$$

    Recall, $n!$ is the factorial of $n$, defined by:

    $$n! = n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1$$

    So for example, the fourth term in the sum above would be $4 \cdot 4! = 4 \cdot (4 \cdot 3 \cdot 2 \cdot 1) = 96$.

  4. Use induction to prove that $7 \mid 13^n - 6^n$ for all positive integers $n$.

  5. Find two primitive Pythagorean triples $a$, $b$, and $c$ such that $a^2 +b^2 = c^2$, $a$ is odd, and $b+c = 81$.  

  6. A large version of the numbered grid shown below is drawn in chalk in a parking lot. 16 people arrange themselves, one person per square on the large grid. At the sound of a bell, if a person is standing on square $n$, they now move to the square whose number is congruent to $5n \pmod{16}$. How many bells must ring before everyone returns to their original squares?  

  7. Find all integers $m$ and $n$ with greatest common divisor $18$ and least common multiple $720$.  

  8. Use the Euclidean Algorithm to find the greatest common divisor of the $12^{th}$ and $15^{th}$ Fibonacci numbers. For the purposes of counting, you may assume the first and second Fibonacci numbers are both equal to one.  

  9. Find all integer solutions to $153x+28y=1$  

  10. Solve $28x \equiv 30 \pmod{46}$  

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