What are the values of the digits a and b, respectively, in the number a 5523879 b, if it is divisible by both a and b :

Options

A. (8, 6)
B. (7, 2)
C. (8, 1)
D. not unique

Answer

To solve the problem of finding the values of the digits \( a \) and \( b \) in the number \( a5523879b \) such that it is divisible by both \( a \) and \( b \), we can follow these steps:

### Step 1: Understand the problem
We need to find digits \( a \) and \( b \) such that the number \( a5523879b \) is divisible by both \( a \) and \( b \). The options given are (8, 6), (7, 2), (8, 1), and "not unique".

### Step 2: Evaluate each option
We will evaluate each option to see if the number formed is divisible by both digits.

#### Option 1: \( a = 8 \), \( b = 6 \)
- The number becomes \( 855238796 \).
- To check divisibility by \( 8 \), we look at the last three digits: \( 796 \).
- \( 796 \div 8 = 99.5 \) (not divisible).
- Therefore, this option is not valid.

#### Option 2: \( a = 7 \), \( b = 2 \)
- The number becomes \( 755238792 \).
- To check divisibility by \( 7 \):
- \( 755238792 \div 7 = 107891256 \) (exactly divisible).
- To check divisibility by \( 2 \):
- The last digit is \( 2 \) (even), so it is divisible by \( 2 \).
- Therefore, this option is valid.

#### Option 3: \( a = 8 \), \( b = 1 \)
- The number becomes \( 855238791 \).
- To check divisibility by \( 8 \):
- The last three digits are \( 791 \).
- \( 791 \div 8 = 98.875 \) (not divisible).
- Therefore, this option is not valid.

#### Option 4: Not unique
- Since we have found a valid option (7, 2), we can conclude that the answer is unique.

### Conclusion
The values of the digits \( a \) and \( b \) are \( 7 \) and \( 2 \), respectively.