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  • Number Puzzle

     Puzzle Prime updated 4 weeks ago 2 Members · 4 Posts
  • Vivek

    Member
    March 9, 2021 at 6:05 pm

    Find all positive integers (n) such that n/d(n) is the prime number, Where d(n) is the number of divisors of n. For example, d(15)=4 {1,3,5,15}.

  • Puzzle Prime

    Administrator
    March 13, 2021 at 12:27 pm

    Hey Vivek,

    Thanks for the fun problem.

    We decompose n = p₁^(a₁)p₂^(a₂)…pₙ^(aₙ) and assume that p₁=n/d(n). Then, we get:

    p₁^(a₁ – 1)p₂^(a₂)…pₙ^(aₙ) = (a₁ + 1)(a+ 1)…(aₙ + 1).

    Now, we note that p^(a) ≥ (a+1) with equality only if p = 2 and a = 1.

    Then, we must have:

    1. a₁ = 3 ⇒ n = 8; (easy)
    2. a₁ = 2 ⇒ n = 9, 12, 18; (easy)
    3. a₁ = 1 ⇒ n = pm, p₁ > 2; Then, aᵢ + 1=2 and 2/n. p₂ = 2 and we get n = 8p, p > 2 or n = 12p, p > 3.

    I just brushed over the case analysis and I hope I haven’t missed some solutions… Even though I suspect I have 😀

  • Vivek

    Member
    March 13, 2021 at 2:58 pm

    Fantastic, Artur. 🙂 You didn’t miss a thing. A minor typo though – first one is 8, not 4. So, 8,9,12,18,8p,12p

    • Puzzle Prime

      Administrator
      March 13, 2021 at 3:38 pm

      Oops, thanks, fixed:)

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