55rd Ukrainian National Mathematical Olympiad - Third Round

Question

In January, Petro used to buy from one to three toy cars every day. On February 1, he tried to make a rectangle of all his cars. When he arranged them into rows of 7 cars, one car remained. When he arranged the cars into rows of 10, there were 2 excessive cars. Can Petro arrange them into rows of 4 cars?

Answer: yes.

Accepted Answer

For some positive integers n, k we have: 7k+1=10n+2 or 7k=10n+1. Therefore, we need to find a number between 29 and 92 that ends with 1 and is divisible by 7. The minimal such number is 21, which is less than 30. The next one is 91. It's easy to check that no other number with these properties exists in the given interval. Therefore, Petro has 7k+1=91+1=92 cars. Since 92 = 4 23, he can arrange them into 4 columns.