Prove that the product of two numbers that take the form $5k+3$, where $k$ is some integer, takes the form $5k+4$.
Prove the irrationality of the following values:
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 $4 \cdot 4! = 4 \cdot (4 \cdot 3 \cdot 2 \cdot 1) = 96$.
Use induction to prove that $7 \mid 13^n - 6^n$ for all positive integers $n$.
Find two primitive Pythagorean triples $a$, $b$, and $c$ such that $a^2 +b^2 = c^2$, $a$ is odd, and $b+c = 81$.
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?
Find all integers $m$ and $n$ with greatest common divisor $18$ and least common multiple $720$.
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.