Question 24
There are some cards on the table. Each card has an integer number written on it. Beto performs the following operation many times: he picks two cards from the table, computes the difference between the numbers that are written on them, he writes this difference on his notebook and then removes those two cards from the table. He can do this operation as many times as he wants, as long as there are at least 2 cards on the table.
In the end, Beto computes the product of all the numbers written on his notebook. Beto's goal is that this product is divisible by $7^{100}$.
a. Prove that if there are initially 207 cards on the table then Beto can always achieve his goal, regardless of which numbers are written on the cards.
b. If there are initially 128 cards on the table, is it true that Beto can always achieve his goal?
In the end, Beto computes the product of all the numbers written on his notebook. Beto's goal is that this product is divisible by $7^{100}$.
a. Prove that if there are initially 207 cards on the table then Beto can always achieve his goal, regardless of which numbers are written on the cards.
b. If there are initially 128 cards on the table, is it true that Beto can always achieve his goal?