If p,q and r are prime numbers such that `r=q+2` and `q=p+2`, then the number of triples of the form `(p,q,r)` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to find the number of triples of the form (p, q, r) where p, q, and r are prime numbers, given the conditions r = q + 2 and q = p + 2.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- We have two equations:
- \( r = q + 2 \)
- \( q = p + 2 \)

2. **Substituting the Values**:
- From the second equation, we can express q in terms of p:
\[ q = p + 2 \]
- Now, substitute this value of q into the first equation:
\[ r = (p + 2) + 2 = p + 4 \]

3. **Identifying the Triples**:
- Now we have:
- \( p \) (a prime number)
- \( q = p + 2 \) (should also be a prime number)
- \( r = p + 4 \) (should also be a prime number)
- Thus, the triplet can be represented as \( (p, p + 2, p + 4) \).

4. **Finding Suitable Prime Numbers**:
- We need to check for values of p such that both \( p + 2 \) and \( p + 4 \) are also prime numbers.
- Let's check small prime numbers for p:
- If \( p = 2 \):
- \( q = 2 + 2 = 4 \) (not prime)
- If \( p = 3 \):
- \( q = 3 + 2 = 5 \) (prime)
- \( r = 3 + 4 = 7 \) (prime)
- So, we have one valid triplet: \( (3, 5, 7) \).
- If \( p = 5 \):
- \( q = 5 + 2 = 7 \) (prime)
- \( r = 5 + 4 = 9 \) (not prime)
- If \( p = 7 \):
- \( q = 7 + 2 = 9 \) (not prime)
- If \( p = 11 \):
- \( q = 11 + 2 = 13 \) (prime)
- \( r = 11 + 4 = 15 \) (not prime)
- Continuing this process, we find that:
- For \( p = 13 \): \( q = 15 \) (not prime)
- For \( p = 17 \): \( q = 19 \) (prime), \( r = 21 \) (not prime)
- For \( p = 19 \): \( q = 21 \) (not prime)

5. **Conclusion**:
- The only valid triplet found is \( (3, 5, 7) \).
- Therefore, the number of triples of the form \( (p, q, r) \) is **1**.

### Final Answer:
The number of triples of the form \( (p, q, r) \) is **1**.