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Difference between revisions of "2025 AIME II Problems/Problem 7"

(Created page with "== Problem == Let <math>A</math> be the set of positive integer divisors of <math>2025</math>. Let <math>B</math> be a randomly selected subset of <math>A</math>. The probabil...")
 
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Let <math>A</math> be the set of positive integer divisors of <math>2025</math>. Let <math>B</math> be a randomly selected subset of <math>A</math>. The probability that <math>B</math> is a nonempty set with the property that the least common multiple of its element is <math>2025</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
Let <math>A</math> be the set of positive integer divisors of <math>2025</math>. Let <math>B</math> be a randomly selected subset of <math>A</math>. The probability that <math>B</math> is a nonempty set with the property that the least common multiple of its element is <math>2025</math> is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
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== Solution ==
+
== Solution 1==
 +
 
 +
We split into different conditions:
 +
 
 +
Note that the numbers in the set need to have a least common multiple of <math>2025</math>, so we need to ensure that the set has at least 1 number that is a multiple of <math>3^4</math> and a number that is a multiple of <math>5^2</math>.
 +
 
 +
Multiples of <math>3^4</math>: <math>81, 405, 2025</math>
 +
 
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Multiples of <math>5^2</math>: <math>25, 75, 225, 675, 2025</math>
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If the set <math>B</math> contains <math>2025</math>, then all of the rest <math>14</math> factors is no longer important. The valid cases are <math>2^{14}</math>.
 +
 
 +
If the set <math>B</math> doesn't contain <math>2025</math>, but contains <math>405</math>, we just need another multiple of <math>5^2</math>. It could be 1 of them, 2 of them, 3 of them, or 4 of them, which has <math>2^4 - 1 = 15</math> cases. Excluding <math>2025, 405, 25, 75, 225, 675,</math> the rest 9 numbers could appear or not appear. Therefore, this case has a valid case of <math>15 \cdot 2^9</math>.
 +
 
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If set <math>B</math> doesn't contain <math>2025</math> nor <math>405</math>, it must contain <math>81</math>. It also needs to contain at least 1 of the multiples from <math>5^2</math>, where it would be <math>15 \cdot 2^8</math>.
 +
 
 +
The total valid cases are <math>2^{14} + 15 \cdot (2^9 + 2^8)</math>, and the total cases are <math>2^{15}</math>.
 +
 
 +
The answer is <math>\cfrac{2^8 \cdot (64 + 30 + 15)}{2^8 \cdot 2^7}= \frac{109}{128}</math>.
 +
 
 +
Desired answer: <math>109 + 128 = \boxed{237}</math>.
 +
 
 +
~ Mitsuihisashi14
 +
 
 +
~ <math>\LaTeX</math> by [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9046]
 +
 
 +
~ Additional edits by [https://artofproblemsolving.com/wiki/index.php/User:Aoum aoum]
 +
 
 +
~ Additional edits by fermat_sLastAMC
 +
 
 +
==Solution 2==
 +
We take the complement and use PIE. Suppose the LCM of the elements of the set is NOT <math>2025</math>. Since <math>2025=3^4 \cdot 5^2</math>, it must be that no element <math>x</math> in the subset satisfies <math>v_3(x)=4</math> OR no element <math>x</math> in the subset satisfies <math>v_5(x)=2</math> (in this case, <math>v_p(n)</math> gives the exponent of <math>p</math> in the prime factorization of <math>n</math>). For the first case, there are <math>4 \cdot 3 = 12</math> possible divisors that could be in our subset (<math>v_3(x)=0,1,2,3,v_5(x)=0,1,2</math> are possible), for a total count of <math>2^{12}</math> subsets. In the second case, there are <math>5 \cdot 2 = 10</math> possible divisors that could be in our subset, for a total count of <math>2^{10}</math> subsets. However, if both conditions are violated, then there are <math>4 \cdot 2 = 8</math> divisors that could be in our subset, for a total count of <math>2^8</math> subsets. Hence, by PIE, the number of subsets whose LCM is NOT <math>2025</math> is equal to <math>2^{12}+2^{10}-2^8</math>. The answer is then <cmath>1-\frac{2^{12}+2^{10}-2^8}{2^{5 \cdot 3}}=\frac{109}{128} \implies \boxed{237}.</cmath>
 +
 
 +
~ [https://artofproblemsolving.com/wiki/index.php/User:Cxsmi cxsmi]
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 +
==Solution 3==
 +
 
 +
Write numbers in the form of <math>3^{a}5^{b}</math> where <math>0\leq a\leq 4; 0\leq b\leq 2</math>
 +
 
 +
There are <math>(4+1)(2+1)=15</math> possible divisors of <math>2025</math>, so the cardinality of the subsets is <math>2^{15}</math>
 +
 
 +
If I select <math>3^4\cdot 5^2</math>, then I guarantee the LCM is 2025, so the other 14 numbers yield <math>2^{14}</math> cases.
 +
 
 +
If I select <math>3^4\cdot 5</math>, then I must select at least one of <math>3^a5^2</math>, but I can select any other <math>9</math> numbers, so there are <cmath>2^9 \left(\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4} \right)=2^9\cdot 15</cmath> ways.
 +
 
 +
If I select <math>3^4</math>, since we can't select <math>3^4\cdot 5</math> or <math>3^4 5^2</math> anymore, there are <math>2^8 \left(\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4} \right)=2^8\cdot 15</math> ways.
 +
 
 +
The answer is then <math>\frac{2^8(15+30+64)}{2^{15}}=\frac{109}{128}\implies \boxed{237}</math>.
 +
 
 +
~ Bluesoul
 +
 
 +
~ Edited by [https://artofproblemsolving.com/wiki/index.php/User:Aoum aoum]
 +
 
 +
~ Additional edits by fermat_sLastAMC
 +
 
 +
==See also==
 +
{{AIME box|year=2025|num-b=6|num-a=8|n=II}}
 +
 
 +
{{MAA Notice}}

Latest revision as of 20:52, 15 February 2025

Problem

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution 1

We split into different conditions:

Note that the numbers in the set need to have a least common multiple of $2025$, so we need to ensure that the set has at least 1 number that is a multiple of $3^4$ and a number that is a multiple of $5^2$.

Multiples of $3^4$: $81, 405, 2025$

Multiples of $5^2$: $25, 75, 225, 675, 2025$

If the set $B$ contains $2025$, then all of the rest $14$ factors is no longer important. The valid cases are $2^{14}$.

If the set $B$ doesn't contain $2025$, but contains $405$, we just need another multiple of $5^2$. It could be 1 of them, 2 of them, 3 of them, or 4 of them, which has $2^4 - 1 = 15$ cases. Excluding $2025, 405, 25, 75, 225, 675,$ the rest 9 numbers could appear or not appear. Therefore, this case has a valid case of $15 \cdot 2^9$.

If set $B$ doesn't contain $2025$ nor $405$, it must contain $81$. It also needs to contain at least 1 of the multiples from $5^2$, where it would be $15 \cdot 2^8$.

The total valid cases are $2^{14} + 15 \cdot (2^9 + 2^8)$, and the total cases are $2^{15}$.

The answer is $\cfrac{2^8 \cdot (64 + 30 + 15)}{2^8 \cdot 2^7}= \frac{109}{128}$.

Desired answer: $109 + 128 = \boxed{237}$.

~ Mitsuihisashi14

~ $\LaTeX$ by eevee9046

~ Additional edits by aoum

~ Additional edits by fermat_sLastAMC

Solution 2

We take the complement and use PIE. Suppose the LCM of the elements of the set is NOT $2025$. Since $2025=3^4 \cdot 5^2$, it must be that no element $x$ in the subset satisfies $v_3(x)=4$ OR no element $x$ in the subset satisfies $v_5(x)=2$ (in this case, $v_p(n)$ gives the exponent of $p$ in the prime factorization of $n$). For the first case, there are $4 \cdot 3 = 12$ possible divisors that could be in our subset ($v_3(x)=0,1,2,3,v_5(x)=0,1,2$ are possible), for a total count of $2^{12}$ subsets. In the second case, there are $5 \cdot 2 = 10$ possible divisors that could be in our subset, for a total count of $2^{10}$ subsets. However, if both conditions are violated, then there are $4 \cdot 2 = 8$ divisors that could be in our subset, for a total count of $2^8$ subsets. Hence, by PIE, the number of subsets whose LCM is NOT $2025$ is equal to $2^{12}+2^{10}-2^8$. The answer is then \[1-\frac{2^{12}+2^{10}-2^8}{2^{5 \cdot 3}}=\frac{109}{128} \implies \boxed{237}.\]

~ cxsmi

Solution 3

Write numbers in the form of $3^{a}5^{b}$ where $0\leq a\leq 4; 0\leq b\leq 2$

There are $(4+1)(2+1)=15$ possible divisors of $2025$, so the cardinality of the subsets is $2^{15}$

If I select $3^4\cdot 5^2$, then I guarantee the LCM is 2025, so the other 14 numbers yield $2^{14}$ cases.

If I select $3^4\cdot 5$, then I must select at least one of $3^a5^2$, but I can select any other $9$ numbers, so there are \[2^9 \left(\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4} \right)=2^9\cdot 15\] ways.

If I select $3^4$, since we can't select $3^4\cdot 5$ or $3^4 5^2$ anymore, there are $2^8 \left(\binom{4}{1}+\binom{4}{2}+\binom{4}{3}+\binom{4}{4} \right)=2^8\cdot 15$ ways.

The answer is then $\frac{2^8(15+30+64)}{2^{15}}=\frac{109}{128}\implies \boxed{237}$.

~ Bluesoul

~ Edited by aoum

~ Additional edits by fermat_sLastAMC

See also

2025 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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