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Difference between revisions of "1995 AHSME Problems/Problem 29"

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==Problem==
 
==Problem==
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For how many three-element sets of positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?
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For how many three-element sets of distinct positive integers <math>\{a,b,c\}</math> is it true that <math>a \times b \times c = 2310</math>?
  
  
 
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 }  </math>
 
<math> \mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 }  </math>
  
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==Solution==
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==Solution 1==
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<math>2310 = 2\cdot 3\cdot 5\cdot 7\cdot 11</math>. The number of ordered triples <math>(x,y,z)</math> with <math>xyz = 2310</math> is therefore <math>3^5</math>, since each prime dividing 2310 divides exactly one of <math>x,y,z</math>.
 
  
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Three of these triples have two of <math>x,y,z</math> equal (namely when one is 2310 and the other two are 1). So there are <math>3^5 - 3</math> with <math>x,y,z</math> distinct.
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<math>2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11</math>. We wish to figure out the number of ways to distribute these prime factors amongst 3 different integers, without over counting triples which are simply permutations of one another.
  
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The number of sets of distinct integers <math>\{ a,b,c\}</math> such that <math>abc = 2310</math> is therefore <math>\frac {3^5 - 3}{6}</math> (accounting for rearrangement), or <math>\boxed{40}</math>.
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We can account for permutations by assuming WLOG that <math>a</math> contains the prime factor 2. Thus, there are <math>3^4</math> ways to position the other 4 prime numbers. Note that, with the exception of when all of the prime factors belong to <math>a</math>, we have over counted each case twice, as for when we put certain prime factors into <math>b</math> and the rest into <math>c</math>, we count the exact same case when we put those prime factors which were in <math>b</math> into <math>c</math>.
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Thus, our total number of cases is <math>\frac{3^4 - 1}{2} = 40 \Rightarrow \boxed{C}.</math>
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==Solution 2==
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The prime factorization of <math>2310</math> is <math>2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11.</math> Therefore, we have the equation <cmath> abc = 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11,</cmath>where <math>a, b, c</math> must be distinct positive integers and order does not matter. There are <math>3</math> ways to assign each prime number on the right-hand side to one of the variables <math>a, b, c,</math> which gives <math>3^5 = 243</math> solutions for <math>(a, b, c).</math> However, three of these solutions have two <math>1</math>s and one <math>2310,</math> which contradicts the fact that <math>a, b, c</math> must be distinct. Because each prime factor appears only once, all other solutions have <math>a, b, c</math> distinct. Correcting for this, we get <math>243 - 3 = 240</math> ordered triples <math>(a, b, c)</math> where <math>a, b, c</math> are all distinct.
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Finally, since order does not matter, we must divide by <math>3!,</math> the number of ways to order <math>a, b, c.</math> This gives the final answer, <cmath>\frac{240}{3!} = \frac{240}{6} = \boxed{40}.</cmath>
  
 
==See also==
 
==See also==
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{{Old AMC12 box|year=1995|num-b=28|num-a=30}}
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{{AHSME box|year=1995|num-b=28|num-a=30}}
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{{MAA Notice}}

Latest revision as of 13:33, 15 January 2023

Problem

For how many three-element sets of distinct positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?


$\mathrm{(A) \ 32 } \qquad \mathrm{(B) \ 36 } \qquad \mathrm{(C) \ 40 } \qquad \mathrm{(D) \ 43 } \qquad \mathrm{(E) \ 45 }$

Solution 1

$2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$. We wish to figure out the number of ways to distribute these prime factors amongst 3 different integers, without over counting triples which are simply permutations of one another.

We can account for permutations by assuming WLOG that $a$ contains the prime factor 2. Thus, there are $3^4$ ways to position the other 4 prime numbers. Note that, with the exception of when all of the prime factors belong to $a$, we have over counted each case twice, as for when we put certain prime factors into $b$ and the rest into $c$, we count the exact same case when we put those prime factors which were in $b$ into $c$.

Thus, our total number of cases is $\frac{3^4 - 1}{2} = 40 \Rightarrow \boxed{C}.$

Solution 2

The prime factorization of $2310$ is $2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11.$ Therefore, we have the equation \[abc = 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11,\]where $a, b, c$ must be distinct positive integers and order does not matter. There are $3$ ways to assign each prime number on the right-hand side to one of the variables $a, b, c,$ which gives $3^5 = 243$ solutions for $(a, b, c).$ However, three of these solutions have two $1$s and one $2310,$ which contradicts the fact that $a, b, c$ must be distinct. Because each prime factor appears only once, all other solutions have $a, b, c$ distinct. Correcting for this, we get $243 - 3 = 240$ ordered triples $(a, b, c)$ where $a, b, c$ are all distinct.

Finally, since order does not matter, we must divide by $3!,$ the number of ways to order $a, b, c.$ This gives the final answer, \[\frac{240}{3!} = \frac{240}{6} = \boxed{40}.\]

See also

1995 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 28 		Followed by
Problem 30
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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