Problems

Prove the divisibility rule for $3$: the number is divisible by $3$ if and only if the sum of its digits is divisible by $3$.

What is a remainder in division by $3$ of the sum $1+2+\cdots +2018$?

What is a remainder in division by $3$ of the number $8^{2019}$?

Show that a number $3333333333332$ is not a perfect square (without using a calculator).

What is a remainder in division by $3$ of the number $5^{21}+{17}^6\times 7^{2019}$?

Show that the sum of any three consecutive integers is divisible by $3$.

In a country far far away, there are only two types of coins: 1 crown and 3 crowns coins. Molly had a bag with only 3 crown coins in it. She used some of these coins to buy herself hat and she got one 1 crown coin back. The next day, all of her friends were jealous of her hat, so she decided to buy identical hats for them. She again only had 3 crown coins in her purse, and she used them to pay for 7 hats. Show that she got a single 1 crown coin back.

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?