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# Determining the Base

Reference > Mathematics > Number Theory > Bases

What 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:

(2n + 2) + (1n + 3) = (1n2 +0n + 1)

n2 - 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) = 3n2 + 3
n2 + 9n + 8 = 3n2 + 3
2n2 - 9n - 5 = 0
(n - 5)(2n + 1) = 0

So n = 5 or -
1
2
. And although fractional bases are possible, they're also outside what we're studying here, so the answer must be 5.

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!

## Questions

1.
In base n, 17 + 4 = 22. Find n.
2.
In base n, 23 x 5 = 203. Find n.
3.
In base n, 200 - 10 = 120. Find n.
4.
In base n, 1 + 11 = 100. Find n.
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Operations in other Bases
Why Use Bases?

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