jmc

Question

In a solar system of n planets, Zorn the World Conqueror can invade m planets at a time, but once there are less than m free worlds left, he stops. If he invades 13 at a time then there are 6 left, and if he invades 14 at a time then there are 5 left. If this solar system has more than 100 planets, what is the smallest number of planets it could have?

Accepted Answer

The given information translates to the congruences align* &n 613, &n 514. align*From the first congruence we obtain that n = 6 + 13k for some integer k. Combining this result with the second congruence, we have 6+13k = n 5 14. Therefore, k 1 14. So, k = 1+14t for some integer t. Substituting 1+14t for k, we have align* n&=6+13k &=6+13(1+14t) &=19+182t 19182. align*The smallest such n greater than 100 is 201.