John cut a piece of paper into 7 smaller pieces. Keeping all of his pieces of paper, he then cut one of them into 13 parts. He continued to cut pieces of paper, sometimes into 7 and sometimes into 13 new slices. After a while he stopped. He claimed that he had exactly 2000 bits of paper. Is it possible that he was right?

On a blackboard, we have the following numbers: $1,2,3,4,\ldots,5000$ (i.e., the first 5000 natural numbers). At each step we choose two numbers, erase them and write the positive difference of them on the blackboard. We continue this process until we have only one number on the blackboard. Can this number be 2015?

Easier variant: try the same starting with the numbers $1,2,3,4,\ldots,12$; can you get 5?

Consider the number $100!=1\cdot 2\cdot 3\cdots 99\cdot 100$. How many zeros are there at the end of it?

We have 100 pennies in a line, with heads up. At the first step, we overturn all the pennies. At the second step, we overturn every second penny. At the third step, we overturn every third penny. We continue and at the $100^{\text{th}}$ step, we overturn the $100^{\text{th}}$ penny. When we stop, how many heads we can see?

Prove that if the number $ABCDE$ is divisible by 271, then $EABCD$ is divisible by 271 as well.