If p & q are relatively prime number in such a way p + q = 10 & p `lt` q . How many pair are possible of p,q.

Options

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

Answer

To solve the problem, we need to find pairs of relatively prime numbers \( p \) and \( q \) such that:

1. \( p + q = 10 \)
2. \( p < q \)

### Step-by-Step Solution:

**Step 1: Identify the range of numbers.**
Since \( p + q = 10 \), both \( p \) and \( q \) must be less than 10. Therefore, we will consider numbers from 1 to 9.

**Hint:** Remember that both \( p \) and \( q \) must be positive integers.

---

**Step 2: List the prime numbers between 1 and 10.**
The prime numbers in this range are: 2, 3, 5, 7.

**Hint:** A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

---

**Step 3: Find pairs of prime numbers that sum to 10.**
We will check combinations of the prime numbers listed:

- \( 2 + 8 \) (8 is not prime)
- \( 3 + 7 \) (both are prime)
- \( 5 + 5 \) (not valid since they are not distinct)
- \( 7 + 3 \) (already counted as \( 3 + 7 \))

The only valid pair of prime numbers that sum to 10 is \( (3, 7) \).

**Hint:** Make sure to check that both numbers in the pair are prime.

---

**Step 4: Check the condition \( p < q \).**
In the pair \( (3, 7) \), we see that \( 3 < 7 \), which satisfies the condition \( p < q \).

**Hint:** Always verify that the order of the pair meets the specified conditions.

---

**Step 5: Conclusion.**
Since the only pair that satisfies both conditions is \( (3, 7) \), we conclude that there is only one possible pair of \( (p, q) \).

**Final Answer:** There is 1 pair possible: \( (3, 7) \).

---