Question

when $555 {10}$ is expressed in this base, it has 4 digits, in the form abab, where a and b are two different digits. what base is it?

Answer 1

In some base
`B`, we have

`abab_B \equiv 555_(10)`

so that

`aB^3 + bB^2 + aB + b = 555`

Factorize the left side.

`aB (B^2 + 1) + b (B^2 + 1) = 555`

`(aB + b) (B^2 + 1) = 555`

Now,
`B` is a positive integer greater than 1, and
`a,b` are positive integers taken from
`\{0,1,2,\ldots,B-1\}`. So consider the prime factorization of 555 on the right side.

`(aB + b) (B^2 + 1) = 3*5*37`

which we can write as

`(aB + b) (B^2 + 1) = 15*(6^2 + 1) = (2*6 + 3) * (6^2+1)`

and so the base is 6, and the number in question is 2323₆.