Assume you start with the minimum possible value for each number so you can only multiply a number by 2 till it reaches its maximum possible value.

If there is a better solution than the current one, then it must have either its maximum value less than the current maximum value, or the minimum value larger than the current minimum value.

Since that we only increase numbers (multiply them by 2), we cannot decrease the current maximum value, so we must multiply the current minimum number by 2.