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?

$$\begin{array}{c|c|c|c} 1&2&3&4\\\hline 5&6&7&8\\\hline 9&10&11&12\\\hline 13&14&15&16 \end{array}$$
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 $28x \equiv 30 \pmod{46}$

◆ ◆ ◆