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

Step 1: Understanding the Concept:
\(aN\) represents the set of all multiples of \(a\). Intersection of sets of multiples gives the set of multiples of the LCM.

Step 2: Detailed Explanation:
\(bN\) = set of multiples of \(b\), i.e., \(\{b, 2b, 3b, \dots\}\). \(cN\) = set of multiples of \(c\). Their intersection \(bN \cap cN\) is the set of numbers that are multiples of both \(b\) and \(c\), i.e., multiples of \(\text{LCM}(b,c)\). Since \(b\) and \(c\) are coprime, \(\text{LCM}(b,c) = bc\). Thus, \(bN \cap cN = (bc)N\). Given that \(bN \cap cN = dN\), we have \((bc)N = dN\), which implies \(d = bc\).

Step 3: Final Answer:
\(d = bc\), which corresponds to option (C).