## Task 14/73

Let $a$ and $b$ each be one of the ten digits $0 ; 1 ; \ldots ; 9$. How many different numbers of the form $a b a b a b$ are possible? Which prime numbers are divisors of all these numbers?
Since for each of the two variables $a$ and $b$ ten different digits can be inserted, there are a total of $10^{2}=100$ different possibilities (number of variations of $n=10$ elements of the 2nd class with repetition).

Among these is also the number 000 000. Apart from this trivial case, the smallest of these numbers is the number 010 101; all other numbers are multiples of this number, and thus also contain the prime factors contained in it. The prime factorization of 010101 yields

$$
010101=3 \cdot 7 \cdot 13 \cdot 37
$$

The non-trivial numbers thus all contain the prime numbers $3 ; 7 ; 13 ; 37$ as divisors.