Question:
The intersection of the subgroups 2Z, 3Z and 5Z, i.e., $2Z \cap 3Z \cap 5Z$ of (Z, +) is
The intersection of the subgroups 2Z, 3Z and 5Z, i.e., $2Z \cap 3Z \cap 5Z$ of (Z, +) is
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Intersection of $mZ$ and $nZ$ is always $lcm(m, n)Z$.
Updated On: Jul 3, 2026
- 6Z
- 15Z
- 10Z
- 30Z
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The intersection of subgroups $aZ, bZ, cZ$ is a subgroup $kZ$ where $k = lcm(a, b, c)$.
Step 2: Meaning
Elements in $2Z \cap 3Z \cap 5Z$ must be multiples of 2, 3, and 5 simultaneously.
Step 3: Analysis
The smallest positive integer divisible by 2, 3, and 5 is $lcm(2, 3, 5) = 2 \times 3 \times 5 = 30$.
Step 4: Conclusion
Therefore, the intersection is the set of all multiples of 30, which is $30Z$. Final Answer: (D)
The intersection of subgroups $aZ, bZ, cZ$ is a subgroup $kZ$ where $k = lcm(a, b, c)$.
Step 2: Meaning
Elements in $2Z \cap 3Z \cap 5Z$ must be multiples of 2, 3, and 5 simultaneously.
Step 3: Analysis
The smallest positive integer divisible by 2, 3, and 5 is $lcm(2, 3, 5) = 2 \times 3 \times 5 = 30$.
Step 4: Conclusion
Therefore, the intersection is the set of all multiples of 30, which is $30Z$. Final Answer: (D)
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