if a N=`{a x:x iin N} and b N cap c N ,` where b,c in N`are relatively prime , then

Options

A. `d=bc`
B. `c=bd`
C. `b=cd`
D. `a=bcd`

Answer

To solve the problem, we need to analyze the given sets and their relationships. We are given:

- \( N = \{ x : x \in \mathbb{N} \} \)
- \( b, c \in N \) are relatively prime.
- \( bN \cap cN = dN \)

We need to find the relationship between \( b, c, d \).

### Step-by-Step Solution:

1. **Understanding the Sets**:
- The set \( bN \) consists of all multiples of \( b \): \( \{ b, 2b, 3b, \ldots \} \).
- The set \( cN \) consists of all multiples of \( c \): \( \{ c, 2c, 3c, \ldots \} \).
- The intersection \( bN \cap cN \) consists of all common multiples of \( b \) and \( c \).

2. **Finding the Intersection**:
- Since \( b \) and \( c \) are relatively prime, they do not share any common factors other than 1.
- The least common multiple (LCM) of two relatively prime numbers \( b \) and \( c \) is given by:
\[ \text{lcm}(b, c) = b \times c \]
- Therefore, the intersection \( bN \cap cN \) can be expressed as:
\[ bN \cap cN = \text{lcm}(b, c)N = (b \times c)N \]

3. **Identifying \( d \)**:
- From the above, we can conclude that:
\[ dN = (b \times c)N \]
- This implies that \( d = b \times c \).

### Conclusion:
Thus, the relationship between \( b, c, \) and \( d \) is:
\[ d = b \times c \]