Triskaidekaphilia (Part II)

6. 13 is a prime such that for any digits a,b > 0 and any integers n,m > 1 one can build an integer divisible by it having exactly n digits equal to a, exactly m digits equal to b and exactly 2 digits equal to 0. Moreover this m+n+2 digit long number is non-trivial, in that none of the two 0 digits is located at the end of the number.

Indeed this is true since:
$13\mid 10\underset{ktimes1}{\underbrace{1\dots 1}}01,\mathrm{\forall }k\in \mathbf{N}$

One other prime with this property is 7. This can easily e extended to any number of digits.

7. 13 is a prime with the property that:

$\mathrm{\forall }a,b\in \mathbf{N}\{\begin{array}{l}p\mid \overline{ab}\\ p\mid \overline{bc}\end{array}\to p\mid \overline{ca}$

This is equivalent to the fact that the following equation system in a, b and c:

$\{\begin{array}{ccccccc}10\cdot a& +& 1\cdot b& +& 0\cdot c& =& 0\\ 0\cdot a& +& 10\cdot b& +& 1\cdot c& =& 0\\ 1\cdot a& +& 0\cdot b& +& 10\cdot c& =& 0\end{array}$

has non-trivial solutions modulo p, hence for its discriminant we have that:

$\left|\begin{array}{ccc}10& 1& 0\\ 0& 10& 1\\ 1& 0& 10\end{array}\right|=1001\equiv 0\mathrm{mod}p$

Therefore the primes with this property are 7, 11 and 13.

8. 13 is a prime with the property that:

$\mathrm{\forall }a,b\in \mathbf{N}\{\begin{array}{l}p\mid \overline{ab}\\ p\mid \overline{bc}\end{array}\to p\mid a+b+c$

9. 13 is a prime with the property that:

$\mathrm{\forall }a,b\in \mathbf{N}\{\begin{array}{l}p\mid \overline{ab}\\ p\mid \overline{bc}\end{array}\to p\mid a^2+b^2+c^2$