SOLUTION: Linae, the Kougra from Shenkuu, went shopping for Kou-jong tiles at the local toy shop the other day. The kou-jong tiles were on sale for only 28 Neopoints each. However, she g

Question 53377: Linae, the Kougra from Shenkuu, went shopping for Kou-jong tiles at the local toy shop the other day. The kou-jong tiles were on sale for only 28 Neopoints each.
However, she got it in her head that, for good luck, the total amount of Neopoints she must spend must be a number where all of the digits are the same.
With that in mind, what is the minimum number of tiles she must buy?

Please help, I cant figure this out!

Found 2 solutions by Nate, rapaljer:
Answer by Nate(3500) (Show Source):
You can put this solution on YOUR website!
Just Do This:
11 / 28 = integer? no
111 / 28 = integer? no
444444 / 28 = integer? yes
15,873

Answer by rapaljer(4551) (Show Source):
You can put this solution on YOUR website!
I think this is a really neat problem. Let x = number of tiles purchased. Then the total cost of the tiles would be 28x, which must be the same digit repeating. The total cost will be
28x = k, or 28x = kk, or 28x = kkk, or 28x = kkkk, etc.

Therefore, you must find a number, such that
x = k/28, or kk/28, or kkk/28, or kkkk/28, etc.

This can also be written as
or or or etc.

The problem then is to find a number with repeating digits that is evenly divisible by 7 and 4. Moreover, since the number must also be divisible by 4, it means that the value of k must be 4 times the number 1 or 11 or 111 or 1111, where you are looking for the smallest such number that is divisible by 7.

So try dividing 11/7, which does not go evenly.
Try 111/7, which does not go evenly.
Likewise continue with, 1111/7, 11111/7, 111111/7, etc. until you find a number that is divisible by 7. The latter divides evenly.

The solution to the problem is 15873 tiles, whose cost is 15873*28 = 444444.

R^2 at SCC