Math CentralQuandaries & Queries


Question from Jenna, a student:

In what base does 25+25=51?

Jenna, we have two responses for you.

Hi Jenna.

The rightmost digit is always the units digit and the digit to its left is the base digit.

For example, in normal decimal notation, 84 has a 4 in the units digit and an 8 in the "base 10" digit. So 84 = 8(10) + 1.

So if we don't know the base, then we can rewrite your question using
B for the base:
25 + 25 = 51
(2B + 5) + (2B + 5) = (5B + 1)

Can you solve for B? If it doesn't come out to a natural number greater than 1, then it isn't a valid base, since we can only have positive whole numbers 2 and up for bases.

Stephen La Rocque.>

Hi Jenna,

Think about what was done to add these two numbers


The person who added them added the first column, 5 + 5, wrote down a 1 and carried something. Then the second column is 2 + 2 + the carry = 5, and hence the carry was a 1. So in what base would you add 5 and 5 and get 1 with a carry of 1?


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