Difference between revisions of "2019 AIME I Problems/Problem 9"

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==Solution==
 
==Solution==
In order to obtain a sum of 7, we must have:
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In order to obtain a sum of <math>7</math>, we must have:
* either a number with 5 divisors (''a fourth power of a prime'') and a number with 2 divisors (''a prime''), or
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* either a number with <math>5</math> divisors (''a fourth power of a prime'') and a number with <math>2</math> divisors (''a prime''), or
* a number with 4 divisors (''a semiprime or a cube of a prime'') and a number with 3 divisors (''a square of a prime''). (No integer greater than 1 can have fewer than 2 divisors.)
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* a number with <math>4</math> divisors (''a semiprime or a cube of a prime'') and a number with <math>3</math> divisors (''a square of a prime''). (No integer greater than <math>1</math> can have fewer than <math>2</math> 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 <math>3^2</math> with 3 divisors, or a fourth power like <math>2^4</math> with 5 divisors. We then find the smallest such values by hand.
+
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 <math>3^2</math> with <math>3</math> divisors, or a fourth power like <math>2^4</math> with <math>5</math> divisors. We then find the smallest such values by hand.
* <math>2^2</math> has two possibilities: 3 and 4, or 4 and 5. Neither works.
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* <math>2^2</math> has two possibilities: <math>3</math> and <math>4</math> or <math>4</math> and <math>5</math>. Neither works.
* <math>3^2</math> has two possibilities: 8 and 9, or 9 and 10. (8,9) and (9,10) both work.
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* <math>3^2</math> has two possibilities: <math>8</math> and <math>9</math> or <math>9</math> and <math>10</math>. <math>(8,9)</math> and <math>(9,10)</math> both work.
* <math>2^4</math> has two possibilities: 15 and 16, or 16 and 17. Only (16,17) works.
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* <math>2^4</math> has two possibilities: <math>15</math> and <math>16</math> or <math>16</math> and <math>17</math>. Only <math>(16,17)</math> works.
* <math>5^2</math> has two possibilities: 24 and 25, or 25 and 26. Only (25,26) works.
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* <math>5^2</math> has two possibilities: <math>24</math> and <math>25</math> or <math>25</math> and <math>26</math>. Only <math>(25,26)</math> works.
* <math>7^2</math> has two possibilities: 48 and 49, or 49 and 50. Neither works.
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* <math>7^2</math> has two possibilities: <math>48</math> and <math>49</math> or <math>49</math> and <math>50</math>. Neither works.
* <math>3^4</math> has two possibilities: 80 and 81, or 81 and 82. Neither works.
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* <math>3^4</math> has two possibilities: <math>80</math> and <math>81</math> or <math>81</math> and <math>82</math>. Neither works.
* <math>11^2</math> has two possibilities: 120 and 121, or 121 and 122. Only (121,122) works.
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* <math>11^2</math> has two possibilities: <math>120</math> and <math>121</math> or <math>121</math> and <math>122</math>. Only <math>(121,122)</math> works.
* <math>13^2</math> has two possibilities: 168 and 169, or 169 and 170. Neither works.
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* <math>13^2</math> has two possibilities: <math>168</math> and <math>169</math> or <math>169</math> and <math>170</math>. Neither works.
* <math>17^2</math> has two possibilities: 288 and 289, or 289 and 290. Neither works.
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* <math>17^2</math> has two possibilities: <math>288</math> and <math>289</math> or <math>289</math> and <math>290</math>. Neither works.
* <math>19^2</math> has two possibilities: 360 and 361, or 361 and 362. Only (361,362) works.
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* <math>19^2</math> has two possibilities: <math>360</math> and <math>361</math> or <math>361</math> and <math>362</math>. Only <math>(361,362)</math> works.
 
Having computed the working possibilities, we take the sum of the corresponding values of <math>n</math>: <math>8+9+16+25+121+361 = \boxed{540}</math>. ~Kepy.
 
Having computed the working possibilities, we take the sum of the corresponding values of <math>n</math>: <math>8+9+16+25+121+361 = \boxed{540}</math>. ~Kepy.
  
  
Possible improvement: since all primes <math>>2</math> are odd, their fourth powers are odd as well, which cannot be adjacent to any primes because both of the adjacent numbers will be even. Thus, we only need to check 16 for the fourth power case. - mathleticguyyy
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Possible improvement: since all primes <math>>2</math> are odd, their fourth powers are odd as well, which cannot be adjacent to any primes because both of the adjacent numbers will be even. Thus, we only need to check <math>16</math> for the fourth power case. - mathleticguyyy
  
 
Note: Bashing would work for this problem, but it would be very tedious.
 
Note: Bashing would work for this problem, but it would be very tedious.

Revision as of 13:03, 26 December 2019

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 or a cube of a prime) 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.


Possible improvement: since all primes $>2$ are odd, their fourth powers are odd as well, which cannot be adjacent to any primes because both of the adjacent numbers will be even. Thus, we only need to check $16$ for the fourth power case. - mathleticguyyy

Note: Bashing would work for this problem, but it would be very tedious.

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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