Basic Notions.

We distinguish two kinds of natural numbers, prime and composite. A number is composite if it is equal to the product of two smaller natural numbers.For example, $6=2\cdot3$. Otherwise, and if the number is not equal to 1, it is called prime. The number 1is neither prime nor composite.

Methodological Remark. If you are a teacher, you might have noticed that this topic comes with a large amount of theoretical notions, some of them familiar from the school program, other new. At the same time, many of the problems are routine and aim to develop some standard technical skills. These sessions might often be organized in the form of games and contests. You can ask students `` Who can factor this huge number first'', or ``Who can fid the greatest prime divisor first''. It is important to understand, that a number can be generally factored in different ways which brings us to the Fundamental Theorem of the Arithmetic.

Theorem. Any natural number different than 1 can be uniquely represented as a product of prime numbers of increasing order.

Here are couple of examples:

Problem 1. Decompose 420 in prime factors. Show that there are different ways which give the same answer.

Answer.We do it in a several steps. First represent 420 as a product of 2 numbers like $42\cdot10$. Then we do the same with $10=2\cdot5$ and $42=6\cdot7$. Finally, $6=2\cdot3$, and $420=2^2\cdot3\cdot5\cdot7$

Problem 2. Is 2⁹ divisible by 2

Answer.Yes, since 2 is one of the factors of in the decomposition of the given number.

Problem 3. Is 2⁹ divisible by 5

Answer.No, since the decomposition does not contain the prime number 5.