Do not use a calculator for any of these problems.
Do your best to solve them, but if you get stuck and can't find an answer to all parts of a problem, don't worry. We can talk about it together, and you can always come back and try to solve it again later.
Problem 1. For each of the following equations, find at least one integer solution $x,y\in\mathbb Z$, or try to prove that none exists.
For example, an integer solution to the first equation is $(x,y)=(3,-2)$, because $3\cdot 5 - 2\cdot 7 = 1$. (You don't have to solve that one.)
Problem 2. $(1+\sqrt{2})^{10}$ is equal to $3363 + 2378\sqrt{2}$. Calculate the value of $(1-\sqrt{2})^{10}$.
Problem 3. Consider the following process: start with a natural number $n$, add up all of its digits in base $10$, and repeat the process with this new number. For example, if we start with the number $n=14832$, the sequence goes like this: In this case, it ends up repeating the number $9$ forever. Does this process always end with some number repeating itself forever? If so, can you prove it? If you use the starting value then will the sequence eventually repeat, and if so, what number will end up repeating forever?