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Part BCSIR NET June 2025terminating-decimals-need-denominator-2n5m-not-just-2n

Terminating decimals need denominator 2n5m not just 2n

Let A = {p/q∈(0,1):p∈N,q=2np/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n} for some n∈N∪n \in \mathbb{N} \cup {0}, gcd(p, q) = 1}, B = {p/q∈(0,1):p∈N,q=2n5mp/q \in (0, 1) : p \in \mathbb{N}, q = 2^{n}5^{m} for some n,m∈N∪n, m \in \mathbb{N} \cup {0}, gcd(p, q) = 1} and C = {p/q ∈ (0, 1) : p/q has terminating decimal expansion} be subsets of (0, 1). Which of the following statements is true?

  1. A.A ⊊ C and B ⊊ C
  2. B.A ⊊ C ⊊ B
  3. C.A ⊊ B ⊊ C
  4. D.A ⊊ B = C✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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More on this topic

The chapter behind this: Elements of set theory: operations, De Morgan and difference — free to read

From The Real Line › Elements of set theory: operations, De Morgan and difference

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