# Determining the Base

Reference > Mathematics > Number Theory > BasesWhat happens if you don't know the base of the calculation? Can you figure it out? Let's try.

**EXAMPLE**

The following problem was written in base *n*: 22 + 13 = 101. Find *n*.

The way to solve this is to write each of the numbers in our long-hand notation, using *n* to represent the base:

(2*n* + 2) + (1*n* + 3) = (1*n*^{2} +0n + 1)

n^{2} - 3n - 4 = 0

(n - 4)(n + 1) = 0

So n = 4 or n = -1.

And although negative bases are possible, they are outside of what we're studying here, so the answer is 4.

**EXAMPLE**

The following problem was written in base *n*: 18 x 11 = 303.

(n + 8)( n + 1) = 3n^{2} + 3

n^{2} + 9n + 8 = 3n^{2} + 3

2n^{2} - 9n - 5 = 0

(n - 5)(2n + 1) = 0

HOWEVER...this is a trick question! If n was 5, how did we end up with the symbol 8 in our problem? We can only use 0, 1, 2, 3, and 4 in base five! Therefore, the answer is: *no solution!*