Thycotic have put up a Code Challenge.

Below is my heavily LINQ-oriented solution:

public interface IConverter
{
    string Convert(int number);
    int Convert(string number);
    string ConvertUsing(IConverter convertor, string number);
}

public abstract class ConverterBase : IConverter
{
    protected abstract string Chars { get; }

    public string Convert(int number)
    {
        return this.Encode(number)
                   .Aggregate(string.Empty, (s, c) => s + c);
    }

    public int Convert(string number)
    {
        return number.Reverse()
                     .Select((c, i) => Chars.IndexOf(c) * (int)Math.Pow(this.Chars.Length, i))
                     .Sum();
    }

    public string ConvertUsing(IConverter convertor, string number)
    {
        return convertor.Convert(this.Convert(number));
    }

    private IEnumerable<char> Encode(int number)
    {
        var remainder = number % this.Chars.Length;
        if (number - remainder > 0)
        {
            foreach (var power in this.Encode((number - remainder) / this.Chars.Length))
            {
                yield return power;
            }
        }
        yield return this.Chars[remainder];
    }
}

public class Converter : ConverterBase
{
    protected override string Chars
    {
        get
        {
            return "0123456789xyz";
        }
    }
}

public class HexConverter : ConverterBase
{
    protected override string Chars
    {
        get
        {
            return "0123456789abcdef";
        }
    }
}

public class BinaryConverter : ConverterBase
{
    protected override string Chars
    {
        get
        {
            return "01";
        }
    }
}

public class OctalConverter : ConverterBase
{
    protected override string Chars
    {
        get
        {
            return "01234567";
        }
    }
}

Note: The code doesn't handle negative inputs or exceptions.