Question

If \(aN = \{an : n \in N\}\) and \(bN \cap cN = dN\), where \(a,b,c \in N\) and \(b,c\) are coprime, then

• A. \(b = cd\)
• B. \(c = bd\)
• C. \(d = bc\)
• D. None of these

Hint:

For coprime numbers, the LCM is the product. The intersection of multiple sets is the set of multiples of the LCM.

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the given conditions and derive the value of \( d \) in the context of sets and number theory.

1. Understanding the Expressions:
   • The expression \( aN = \{an : n \in N\} \) represents a set of multiples of \( a \).
   • Similarly, \( bN \) and \( cN \) are sets representing multiples of \( b \) and \( c \) respectively.
   • The intersection \( bN \cap cN \) represents the set of numbers that are common multiples of both \( b \) and \( c \).
   • This common set is given by \( dN \), which indicates that the common multiples are also multiples of \( d \).
2. Using the Coprime Condition:
   • Given that \( b \) and \( c \) are coprime, it means \(\gcd(b, c) = 1\).
   • The least common multiple (LCM) of two coprime numbers is equal to their product, \(\mathrm{lcm}(b, c) = b \times c\).
3. Analyzing \( dN = bN \cap cN \):
   • Since \(\mathrm{lcm}(b, c) = b \times c\), the least common multiple describes the smallest positive integer that is in both \( bN \) and \( cN \).
   • Therefore, the intersection \( bN \cap cN \) is equal to the multiples of the least common multiple, which means \( dN = \mathrm{lcm}(b, c)N= (b \times c)N\).
   • Hence, we conclude that \( d = bc \).
4. Conclusion:
   • The correct answer is \( d = bc \).
   • The other options, \( b = cd \) and \( c = bd \), do not satisfy the condition based on the properties of coprime numbers and LCM.