Problems

A natural number is divisible by $5$ and by $7$. Should it be divisible by $35$?

The number $A$ is not divisible by 3. Is it possible that the number $2A$ is divisible by 3?

List the first $10$ prime numbers and write the prime decomposition of $2910$.

Lisa knows that $A$ is an even number. But she is not sure if $3A$ is divisible by $6$. What do you think?

George divided number $a$ by number $b$ with the remainder $d$ and the quotient $c$. How will the remainder and the quotient change if the dividend and the divisor are increased by a factor of 3?

Let us introduce the notation – we denote the product of all natural numbers from 1 to $n$ by $n!$. For example, $5!=1\times 2\times 3\times 4\times 5=120$.

a) Prove that the product of any three consecutive natural numbers is divisible by $3!=6$.

b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?

Can a sum of three different natural numbers be divisible by each of those numbers?

A young mathematician felt very sad and lonely during New Year’s Eve. The main reason for his sadness (have you guessed already?) was the lack of mathematical problems. So he decided to create a new one on his own. He wrote the following words on a small piece of paper: “Find the smallest natural number $n$ such that $n!$ is divisible by 2025”, but unfortunately he immediately forgot the answer. What is the correct answer to this question?

Find a natural number $n$ such that all the numbers $n+1$, $n+2$, ..., $n+2025$ are composite.

The numbers $2^{2025}$ and $5^{2025}$ are expanded and their digits are written out consecutively on one page. How many digits are on the page?