Number Puzzle

[Vivek]

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]

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)(a₂ + 1)…(aₙ + 1).

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

Then, we must have:

1. a₁ = 3 ⇒ n = 8; (easy)
   
2. a₁ = 2 ⇒ n = 9, 12, 18; (easy)
   
3. a₁ = 1 ⇒ n = p₁m, 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]

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]

Oops, thanks, fixed:)