(a) The numbers $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be $97$?
 
(b) The numbers $1 , 2, 22, 23 , . . . , 210$ are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have? 

(A.Shapovalov)
### Part (a)

1. **Initial Setup**: The numbers on the blackboard are \(1, 2, 4, 8, 16, 32, 64, 128\).

2. **Operation Description**: We are allowed to erase any two numbers and write their difference instead. This operation is repeated until only one number remains.

3. **Objective**: Determine if the final remaining number can be \(97\).

4. **Modulo Analysis**: Consider the numbers modulo \(64\). Since \(97 \equiv 33 \pmod{64}\), we need to check if we can express \(33\) using the given numbers modulo \(64\).

5. **Expression in Terms of Given Numbers**:
   \[
   33 = 128 - 64 - 1 - 2 - 4 - 8 + 16 + 32
   \]

6. **Verification**:
   - Start with the numbers \(1, 2, 4, 8, 16, 32, 64, 128\).
   - Perform the operations to achieve the expression \(128 - 64 - 1 - 2 - 4 - 8 + 16 + 32\).

7. **Step-by-Step Operations**:
   - \(128 - 64 = 64\)
   - \(64 - 32 = 32\)
   - \(32 - 16 = 16\)
   - \(16 - 8 = 8\)
   - \(8 - 4 = 4\)
   - \(4 - 2 = 2\)
   - \(2 - 1 = 1\)
   - Combine the results to get \(128 - 64 - 1 - 2 - 4 - 8 + 16 + 32 = 97\).

Thus, it is possible to obtain \(97\) as the final number.

The final answer is any odd number between \( \boxed{ 1 } \) and \(210\).