Mathematical and Physical Journal
for High Schools
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Problem A. 483. (May 2009)

A. 483. For abritrary integers 0<k\len and a>1, define {\binom{n}{k}}_{a} =
\frac{(a^n-1)(a^{n-1}-1)\cdot \ldots \cdot(a^{n-k+1}-1)}{(a^k-1)(a^{k-1}-1) \cdot \ldots
\cdot(a-1)}.

(a) Show that {\binom{n}{k}}_{a} is an integer.

(b) Do there exist integers 0<k<n<m and a>1, for which {\binom{m}{1}}_{a} divides {\binom{n}{k}}_{a}?

(5 pont)

Deadline expired on June 15, 2009.


Statistics:

5 students sent a solution.
5 points:Nagy 235 János, Tomon István.
2 points:3 students.

Problems in Mathematics of KöMaL, May 2009