Koteawarao,

When you are working in base 7 the only "digits" allowed are 0, 1, 2, 3, 4, 5 and 6. In your answer you have the digit 8 so the answer given can't be correct. The symbol 8 has no meaning in base 7.

To solve your problem you want digits a, b and c so that

abc₇ = cba₉

where a, b and c are either 0, 1, 2, ... ,6. Using the definition of numbers expressed in different bases this can be written

a × 7² + b × 7 + c = c × 9² + b × 9 + a

On simplification this becomes

4(6a - 10c) = b

Hence 4 divided b and since b = 0, 1, ..., 6 either b = 0 or b = 4.

If b = 0 then 6a = 10c. You know that a and c are between 0 and 6 so what values of a and c make 6a = 10c? Check the values you obtained for a, b and c solve the problem.

If b = 4 then 6a - 10c = 1. Check all possible values of a and c. Do you find a solution.

I found two choices for a, b and c that solve the problem.

Harley