Problems

Show that numbers $12n+1$ and $12n+7$ are relatively prime.

If natural numbers $a,b$ and $c$ are lengths of the sides of a right triangle (such that $a^2+b^2=c^2$), show that at least one of these numbers is divisible by $3$.

Tom got a really bad grade from the last test and once he got the test back, he started to tear it up. He is tearing it into little pieces in the following manner: He picks up a piece and tears it into either $4$ or $10$ smaller pieces. Can he eventually have exactly 200,000 pieces?

Show that any natural number has the same remainder when divided by $3$ as the sum of its digits.

Show that $n^3+2n$ is divisible by $3$ for a natural $n$.

Prove that if $a^3-b^3$, for $a$ and $b$ natural, is divisible by $3$, then it is divisible by $9$.

Anna has $20$ novels and $25$ comic books on her shelf. She doesn’t really keep her room very tidy and so she also has a lot of novels and comic books in various places around her room. Each time she reaches for the shelf, she takes two books and puts one back. If she takes two novels or two comic books, she puts a novel back on the shelf. If she takes a novel and a comic book, she places another comic book on the shelf. That way, her shelf sistematically empties. Show that eventually there will be a lone comic book standing on her shelf and all her other books scattered across her room.

A knight in chess moves in an “L” pattern – two squares in one direction and one square in a perpendicular direction. Starting with a knight in the bottom right corner of a regular $8\times 8$ chessboard, can you move it some number of times according to the rules in such a way that it visits every square on the chessboard exactly once and ends up in the top left corner?

Nine lightbulbs are arranged in a $3\times 3$ square. Some are on, some are off. You are allowed to change the state of all the bulbs in a column or in a row. That means all the bulbs in that row or column that were off light up and the ones that were on go dark. Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?

Anna’s sister, Claire, has $10$ novels, $11$ textbooks and $12$ comic books on her shelf. She also doesn’t like to keep all her books there. Each time she takes two books of different type from the shelf and puts a book of the third type back on. So for example, she might take a novel and a comic book and put a textbook back. Show that eventually there is only a single textbook, and exactly a textbook, left on her shelf.