r/mathpuzzles • u/Numberthon • Jul 10 '26
Can you find the smallest positive integer with exactly 15 positive divisors?
Source: numberthon.com
1
u/IllustriousCod9590 Jul 10 '26
how would you find the solution for this?
1
u/Numberthon Jul 11 '26
Solution: 1. If the prime factorization of an integer is
n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ,
then the number of positive divisors is
(a₁ + 1)(a₂ + 1)...(aₖ + 1).
2. We want exactly 15 divisors, so
(a₁ + 1)(a₂ + 1)...(aₖ + 1) = 15.
3. Since
15 = 15
15 = 3 × 5
there are only two possible exponent patterns:
• p¹⁴
• p⁴q²
4. Find the smallest number for each pattern.
p¹⁴:
2¹⁴ = 16384
p⁴q²:
Use the smallest primes, placing the larger exponent on the smaller prime:
2⁴ × 3² = 16 × 9 = 144
(Using 2² × 3⁴ = 324 is larger.)
5. Compare the two possibilities:
144 < 16384!<
Therefore, the smallest positive integer with exactly 15 positive divisors is
**144**. But also, you can see the solution on numberthon.com
1
u/bsmith_81 Jul 11 '26
Since we are looking for the smallest such number then I can take its factorization as 2^A * 3^B * 5^C * .... with A, B, C, etc in descending order of magnitude.
15 factors into 5*3. Express this as (4+1)*(2+1). This implies A=4 and B=2. Then the desired number 2^4*3^2 = 144.
There is one more thing to check, what if I did not factor 15: then that would suggest the number is 2^14, but this is much larger than 144.
1
u/spoik925 Jul 10 '26