Difference between revisions of "2017 AIME I Problems/Problem 12"

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Problem 12

Call a set $S$ product-free if there do not exist $a, b, c \in S$ (not necessarily distinct) such that $a b = c$ . For example, the empty set and the set $\{16, 20\}$ are product-free, whereas the sets $\{4, 16\}$ and $\{2, 8, 16\}$ are not product-free. Find the number of product-free subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ .

Solution 1 (Casework)

We shall solve this problem by doing casework on the lowest element of the subset. Note that the number $1$ cannot be in the subset because $1*1=1$ . Let $S$ be a product-free set. If the lowest element of $S$ is $2$ , we consider the set $\{3, 6, 9\}$ . We see that 5 of these subsets can be a subset of $S$ ( $\{3\}$ , $\{6\}$ , $\{9\}$ , $\{6, 9\}$ , and the empty set). Now consider the set $\{5, 10\}$ . We see that 3 of these subsets can be a subset of $S$ ( $\{5\}$ , $\{10\}$ , and the empty set). Note that $4$ cannot be an element of $S$ , because $2$ is. Now consider the set $\{7, 8\}$ . All four of these subsets can be a subset of $S$ . So if the smallest element of $S$ is $2$ , there are $5*3*4=60$ possible such sets.

If the smallest element of $S$ is $3$ , the only restriction we have is that $9$ is not in $S$ . This leaves us $2^6=64$ such sets.

If the smallest element of $S$ is not $2$ or $3$ , then $S$ can be any subset of $\{4, 5, 6, 7, 8, 9, 10\}$ , including the empty set. This gives us $2^7=128$ such subsets.

So our answer is $60+64+128=\boxed{252}$ .

Solution 2 (PIE)

We will consider the $2^9 = 512$ subsets that do not contain 1. A subset is product-free if and only if it does not contain one of the groups $\{2, 4\}, \{3, 9\}, \{2, 3, 6\},$ or $\{2, 5, 10\}$ . There are $2^7$ subsets that contain 2 and 4 since each of the seven elements other than 2 and 4 can either be in the subset or not. Similarly, there are $2^7$ subsets that contain 3 and 9, $2^6$ subsets that contain 2, 3, and 6, and $2^6$ subsets that contain 2, 5, and 10. The number of sets that contain one of the groups is: $2^7 + 2^7 + 2^6 + 2^6 = 384$ For sets that contain two of the groups, we have: $2^5 + 2^5 + 2^5 + 2^5 + 2^4 + 2^4 = 160$ For sets that contain three of the groups, we have: $2^4 + 2^3 + 2^3 + 2^3 = 40$ For sets that contain all of the groups, we have: $2^2 = 4$ By the principle of inclusion and exclusion, the number of product-free subsets is $512 - (384 - 160 + 40 - 4) = \boxed{252}$ .

Solution 3

Let $X$ be a product-free subset, and note that 1 is not in $x$ . We consider four cases: 1.) both 2 and 3 are not in $X$ . Then there are $2^7=128$ possible subsets for this case. 2.) 2 is in $X$ , but 3 is not. Then 4 in not in $X$ , so there are $2^6=64$ subsets; however, there is a $\frac{1}{4}$ chance that 5 and 10 are both in $X$ , so there are $64\cdot \frac{3}{4}=48$ subsets for this case. 3.) 2 is not in $X$ , but 3 is. Then, 9 is not in $X$ , so there are $2^6=64$ subsets for this case. 4.) 2 and 3 are both in $X$ . Then, 4, 6, and 9 are not in $X$ , so there are $2^4=16$ total subsets; however, there is a $\frac{1}{4}$ chance that 5 and 10 are both in $X$ , so there are $16\cdot \frac{3}{4}=12$ subsets for this case. Hence our answer is $128+48+64+12=\boxed{252}$ .

@@ Line 23: / Line 23: @@
 By the principle of inclusion and exclusion, the number of product-free subsets is
 <cmath>512 - (384 - 160 + 40 - 4) = \boxed{252}</cmath>.
+==Solution 3==
+Let <math>X</math> be a product-free subset, and note that 1 is not in <math>x</math>. We consider four cases: 1.) both 2 and 3 are not in <math>X</math>. Then there are <math>2^7=128</math> possible subsets for this case. 2.) 2 is in <math>X</math>, but 3 is not. Then 4 in not in <math>X</math>, so there are <math>2^6=64</math> subsets; however, there is a <math>\frac{1}{4}</math> chance that 5 and 10 are both in <math>X</math>, so there are <math>64\cdot \frac{3}{4}=48</math> subsets for this case. 3.) 2 is not in <math>X</math>, but 3 is. Then, 9 is not in <math>X</math>, so there are <math>2^6=64</math> subsets for this case. 4.) 2 and 3 are both in <math>X</math>. Then, 4, 6, and 9 are not in <math>X</math>, so there are <math>2^4=16</math> total subsets; however, there is a <math>\frac{1}{4}</math> chance that 5 and 10 are both in <math>X</math>, so there are <math>16\cdot \frac{3}{4}=12</math> subsets for this case.
+Hence our answer is <math>128+48+64+12=\boxed{252}</math>.
 ==See Also==
 {{AIME box|year=2017|n=I|num-b=11|num-a=13}}
 {{MAA Notice}}

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