It's All About the Base

Bessie the cow has been taking computing classes at her local college (or "cow-ledge," in her case), and she has recently become very excited about writing numbers in different bases.

Recall that a number written in base $b$ has digit positions representing 1, b, b^2, b^3, and so on, from right to left. For example, in our familiar base-$10$ numbering system, the digit positions represent $1$'s, $10$'s, $100$'s, $1000$'s, and so on. The sequence of digits $1234$, interpreted in base $10$, means

1(1000) + 2(100) + 3(10) + 4(1)

The same sequence of digits $1234$, interpreted in base $5$, means

1(125) + 2(25) + 3(5) + 4(1)

which evaluates to $194$ in base $10$. Bessie notices that as the base increases, so does the number represented by a fixed sequence of digits. For example, $1234$ in base $7$ represents a larger number than $1234$ in base $6$.

When writing numbers in base $b$, each digit can range from $0$ through b-1. For example, in base $10$, each digit is in the range $0..9$, while in base $5$, each digit is in the range $0..4$.

It is also possible to use bases larger than $10$. Computer scientists often use base $16$ ("hexadecimal"), where the letters $A..F$ represent digit values from $10$ through $15$. For example, $BEEF$ in hexadecimal corresponds to

11(4096) + 14(256) + 14(16) + 15

which evaluates to $48879$ in base $10$.

Bessie is intrigued by the idea of using bases much larger than $10$. She takes a number $n$ and writes it down in two different bases $x$ and $y$, where both $x$ and $y$ are in the range $10..15000$. Interestingly, in both cases she obtains a sequence of exactly $3$ digits, and every digit happens to be in the range $1..9$.

Unfortunately, due to Bessie's poor memory, she has now forgotten $n,x$, and $y$. Given only the two $3$-digit sequences she wrote down, help her determine the two bases $x$ and $y$ that she used.

Note that because $x$ and $y$ can be as large as $15000$, a program that exhaustively tries every possible pair of bases (nearly 15000^2 possibilities) will not run within the time limit and will not receive full credit.