when does one fibonacci number divide another? let $f_0 = 0$, $f_1 = 1$, and for $n\\geq 2$, $f_n = f_{n-1} + f_{n-2}$. it is well known that for $f_m > 1$ this last result was used in yuri matijasevi\\u{c}\'s solution of hilbert\'s 10th problem. using simple combinatorial arguments, we derive previuosly unknown necessary and sufficient conditions for the following question: for any $l \\geq 1$ when does $f_m^l$ divide $f_n$? our method allows us to answer this same question for any lucas sequence of the first kind, defined by $u_0 = 0$, $u_1 = 1$, and for $n\\geq 2$, $u_n = au_{n-1} + bu_{n-2}$. this talk is based on joint work with harvey mudd college undergraduate jeremy rouse, while attending the 10th international conference on applications of fibonacci numbers.