2019 AIME I Problems/Problem 9

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Problem 9

Let $\tau (n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\tau (n) + \tau (n+1) = 7$.

Solution

In order to obtain a sum of 7, we must have:

  • either a number with 5 divisors (a fourth power of a prime) and a number with 2 divisors (a prime), or
  • a number with 4 divisors (a semiprime) and a number with 3 divisors (a square of a prime). (No integer greater than 1 can have fewer than 2 divisors.)

Since both of these cases contain a number with an odd number of divisors, that number must be an even power of a prime. These can come in the form of a square like $3^2$ with 3 divisors, or a fourth power like $2^4$ with 5 divisors. We then find the smallest such values by hand.

  • $2^2$ has two possibilities: 3 and 4, or 4 and 5. Neither works.
  • $3^2$ has two possibilities: 8 and 9, or 9 and 10. (8,9) and (9,10) both work.
  • $2^4$ has two possibilities: 15 and 16, or 16 and 17. Only (16,17) works.
  • $5^2$ has two possibilities: 24 and 25, or 25 and 26. Only (25,26) works.
  • $7^2$ has two possibilities: 48 and 49, or 49 and 50. Neither works.
  • $3^4$ has two possibilities: 80 and 81, or 81 and 82. Neither works.
  • $11^2$ has two possibilities: 120 and 121, or 121 and 122. Only (121,122) works.
  • $13^2$ has two possibilities: 168 and 169, or 169 and 170. Neither works.
  • $17^2$ has two possibilities: 288 and 289, or 289 and 290. Neither works.
  • $19^2$ has two possibilities: 360 and 361, or 361 and 362. Only (361,362) works.

Having computed the working possibilities, we take the sum of the corresponding values of $n$: $8+9+16+25+121+361 = \boxed{540}$. ~Kepy.

See Also

2019 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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