Problems

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

Find three different natural numbers, larger than $100$ such that each of them is divisible by the difference of the other two numbers? The values of differences also have to be different from each other.

There are four numbers written in a row. The first number is $100$. It is known that if we divide the first number by the second number we will get a prime number as a result, if we the second number by the third number we will get a prime number, and if we divide the third number by the fourth number we will also get a prime number. Can all the resulting prime numbers be distinct?

We know that the product $c\times d$ is divisible by a prime $p$. Show that either $c$ or $d$ must be divisible by $p$.

The number $a$ has a prime factorization $2^3\times 3^2\times 7^2\times 11$. Is it divisible by $54$? Is it divisible by $154$?

Two numbers are given in terms of their prime factorizations: $a=2^3\times 3^2\times 5\times {11}^2\times {17}^2$ and $b=2\times 5^3\times 7^2\times 11\times 13$.

a) What is the greatest common divisor $\gcd (a,b)$ of these numbers?

b) What is their least common multiple $\mathrm{lcm}(a,b)$?

c) Write down the prime factorization of $\gcd (a,b)\times \mathrm{lcm}(a,b)$. Then write the prime factorization of $a\times b$. What do you notice?

Find all natural numbers $n$ for which there exist integers $a,b,c$ such that $a+b+c=0$ and the number $a^n+b^n+c^n$ is prime.

Alice the fox and Basilio the cat have grown $20$ counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has $7$ empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Cut a $7\times 7$ square into $9$ rectangles, out of which you can construct any rectangle whose sidelengths are less than $7$. Show how to construct the rectangles.

Find all the prime numbers $p$ such that the number $2p^2+1$ is also prime.