If a,b,and c are positive prime numbers , in the equation a-b=c, either b or c must represent which number ?

Options

A. 13
B. 11
C. 7
D. 5

Answer

To solve the problem, we need to analyze the equation \( a - b = c \) where \( a, b, \) and \( c \) are positive prime numbers. We will test the given options (13, 11, 7, and 5) to see which one can be represented by either \( b \) or \( c \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( a - b = c \) implies that \( a \) must be greater than \( b \) (since \( c \) is positive). Therefore, we will always choose \( a \) to be the larger prime number.

2. **Testing the Options**:
We will check each option to see if it can be represented as \( b \) or \( c \) when \( a \) and \( b \) are both prime numbers.

3. **Testing Option 13**:
- Let \( a = 13 \).
- Possible values for \( b \) (which must be prime and less than 13): 2, 3, 5, 7, 11.
- Testing:
- \( b = 11 \): \( 13 - 11 = 2 \) (valid, \( c = 2 \)).
- \( b = 7 \): \( 13 - 7 = 6 \) (not valid, 6 is not prime).
- \( b = 5 \): \( 13 - 5 = 8 \) (not valid, 8 is not prime).
- \( b = 3 \): \( 13 - 3 = 10 \) (not valid, 10 is not prime).
- \( b = 2 \): \( 13 - 2 = 11 \) (valid, \( c = 11 \)).
- Valid pairs: (13, 11, 2) and (13, 2, 11).

4. **Testing Option 11**:
- Let \( a = 11 \).
- Possible values for \( b \): 2, 3, 5, 7.
- Testing:
- \( b = 7 \): \( 11 - 7 = 4 \) (not valid, 4 is not prime).
- \( b = 5 \): \( 11 - 5 = 6 \) (not valid, 6 is not prime).
- \( b = 3 \): \( 11 - 3 = 8 \) (not valid, 8 is not prime).
- \( b = 2 \): \( 11 - 2 = 9 \) (not valid, 9 is not prime).
- No valid pairs.

5. **Testing Option 7**:
- Let \( a = 7 \).
- Possible values for \( b \): 2, 3, 5.
- Testing:
- \( b = 5 \): \( 7 - 5 = 2 \) (valid, \( c = 2 \)).
- \( b = 3 \): \( 7 - 3 = 4 \) (not valid, 4 is not prime).
- \( b = 2 \): \( 7 - 2 = 5 \) (valid, \( c = 5 \)).
- Valid pairs: (7, 5, 2) and (7, 2, 5).

6. **Testing Option 5**:
- Let \( a = 5 \).
- Possible values for \( b \): 2, 3.
- Testing:
- \( b = 3 \): \( 5 - 3 = 2 \) (valid, \( c = 2 \)).
- \( b = 2 \): \( 5 - 2 = 3 \) (valid, \( c = 3 \)).
- Valid pairs: (5, 3, 2) and (5, 2, 3).

7. **Conclusion**:
After testing all options, we find that:
- 5 can be represented as \( c \) in the pairs (5, 3, 2) and (5, 2, 3).
- 7 can also be represented as \( b \) in the pairs (7, 5, 2) and (7, 2, 5).
- However, 11 does not yield valid pairs.

Thus, the answer is that either \( b \) or \( c \) must represent the number **5**.