Describe (A ∩ B) ∩ C for multiples of 2, 5, and 10: Let A = {multiples of 2 in N}, B = {multiples of 5 in N}, C = {multiples of 10 in N}. Describe (A ∩ B) ∩ C.

Introduction / Context:
A ∩ B is the set of natural numbers that are both multiples of 2 and of 5, i.e., multiples of lcm(2,5) = 10. Intersecting that with C yields C again.

Given Data / Assumptions:

• A = {2,4,6,...}
• B = {5,10,15,...}
• C = {10,20,30,...}

Concept / Approach:
A ∩ B = {multiples of 10}. Then (A ∩ B) ∩ C = {multiples of 10} ∩ {multiples of 10} = {multiples of 10}.

Step-by-Step Solution:
A ∩ B = C(A ∩ B) ∩ C = C ∩ C = C

Verification / Alternative check:
Element test: any k divisible by 10 is automatically divisible by 2 and 5; thus C ⊆ A ∩ B and equality holds.

Why Other Options Are Wrong:
They describe larger sets (A, B) or a redundant expression (A ∩ B) without the final restriction; the exact result is C.

Common Pitfalls:
Forgetting that lcm determines simultaneous divisibility for intersections of multiples.

Final Answer:
C