US-TST 2008 Problem 8

Mr. Fat and Ms. Taf play a game. Mr. Fat chooses a sequence of positive integers $k_1, k_2, \ldots , k_n$ . Ms. Taf must guess this sequence of integers. She is allowed to give Mr. Fat a red card and a blue card, each with an integer written on it. Mr. Fat replaces the number on the red card with $k_1$ times the number on the red card plus the number on the blue card, and replaces the number on the blue card with the number originally on the red card. He repeats this process with number $k_2$ . (That is, he replaces the number on the red card with $k_2$ times the number now on the red card plus the number now on the blue card, and replaces the number on the blue card with the number that was just placed on the red card.) He then repeats this process with each of the numbers $k_3, \ldots k_n$ , in this order. After has has gone through the sequence of integers, Mr. Fat then gives the cards back to Ms. Taf. How many times must Ms. Taf submit the red and blue cards in order to be able to determine the sequence of integers $k_1, k_2, \ldots k_n$ ?