Question

The statement below is correct in at least one number system (besides base-1). That is, the statement is correct if we assume the numbers are expressed in a base other than 10. It is up to you to find out which number base makes each statement correct. You need to justify your answer by converting the numbers in each operation to base 10 and showing that the statement is correct. For example, 36/6 = 7 is clearly not correct in base 10 but it is correct in base 12 because 3612 = 4210 and 4210/610 = 710. Thus, 3612/612 = 712 is true.

25 + 1 + 15 + 229 = 261

Answer 1

The base is 19

Step-by-step explanation:

Given

`25 + 1 + 15 + 229 = 261`

Required

Determine the base

Represent the base with n;

So, we have

`25_n + 1_n + 15_n + 229_n = 261_n`

Convert the above to base 10;

`2 * n^1 + 5 * n^0 + 1 * n^0 + 1 * n^1 + 5 * n^0 + 2 * n^2 + 2 * n^1 + 9 * n^0 = 2 * n^2 + 6 * n^1 + 1 * n^0`

`2 * n + 5 * 1 + 1 * 1 + 1 * n + 5 * 1 + 2 * n^2 + 2 * n + 9 * 1 = 2 * n^2 + 6 * n + 1 * 1`

`2n + 5 + 1 + n + 5 + 2n^2 + 2n + 9 = 2n^2 + 6n + 1`

Collect Like Terms

`2n^2 + 2n + 2n +n+ 5 + 1 + 5 + 9 = 2n^2 + 6n + 1`

`2n^2 +5n + 20 = 2n^2 +6n+1`

Collect Like Terms

`2n^2 - 2n^2 +5n -6n = 1-20`

`-n = -19`

`n = 19`