sub only_1_0_in_base_x {
    my ($num, $base) = @_;
    if ($num / $base != int($num / $base)) {
        $num--;
        return 0 if $num / $base != int($num / $base);
    }
    my $exponent = int(log($num) / log($base));

    my $max = 0;
    $max += $base ** $_ for 1 .. $exponent;

    return 0 if $num > $max;    

    while ($exponent) {
        my $next = $base ** $exponent;
        return 1 if $next == $num;

        $num -= $next if $num > $next;
        $max -= $next;

        return 0 if $num > $max;
    }
}