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Difference between revisions of "2005 AMC 10A Problems/Problem 15"

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How many positive cubes divide <math> 3! \cdot 5! \cdot 7! </math> ?
 
How many positive cubes divide <math> 3! \cdot 5! \cdot 7! </math> ?
  
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<math> \mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6 </math>
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<math> \textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 5\qquad \textbf{(E) } 6 </math>
  
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==Solution==
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== Solution 1 ==
 
<math> 3! \cdot 5! \cdot 7! = (3\cdot2\cdot1) \cdot (5\cdot4\cdot3\cdot2\cdot1) \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) = 2^{8}\cdot3^{4}\cdot5^{2}\cdot7^{1}</math>
 
<math> 3! \cdot 5! \cdot 7! = (3\cdot2\cdot1) \cdot (5\cdot4\cdot3\cdot2\cdot1) \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) = 2^{8}\cdot3^{4}\cdot5^{2}\cdot7^{1}</math>
  
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Therefore, a [[perfect cube]] that divides <math> 3! \cdot 5! \cdot 7! </math> must be in the form <math>2^{a}\cdot3^{b}\cdot5^{c}\cdot7^{d}</math> where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are [[nonnegative]] [[multiple]]s of <math>3</math> that are less than or equal to <math>8</math>, <math>4</math>, <math>2</math> and <math>1</math>, respectively.  
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Therefore, a [[perfect cube]] that divides <math> 3! \cdot 5! \cdot 7! </math> must be in the form <math>2^{a}\cdot3^{b}\cdot5^{c}\cdot7^{d}</math> where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are [[nonnegative]] [[multiple]]s of <math>3</math> that are less than or equal to <math>8, 5, 2</math> and <math>1,</math> respectively.  
  
 
So:  
 
So:  
  
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<math>a\in\{0,3,6\}</math> (<math>3</math> posibilities)
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<math>a\in\{0,3,6\}</math> (<math>3</math> possibilities)
  
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<math>b\in\{0,3\}</math> (<math>2</math> posibilities)
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<math>b\in\{0,3\}</math> (<math>2</math> possibilities)
  
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<math>c\in\{0\}</math> (<math>1</math> posibility)
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<math>c\in\{0\}</math> (<math>1</math> possibility)
  
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<math>d\in\{0\}</math>(<math>1</math> posibility)
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<math>d\in\{0\}</math>(<math>1</math> possibility)
  
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So the number of perfect cubes that divide <math> 3! \cdot 5! \cdot 7! </math> is <math>3\cdot2\cdot1\cdot1 = \boxed{\textbf{(E) }6}</math>
  
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So the number of perfect cubes that divide <math> 3! \cdot 5! \cdot 7! </math> is <math>3\cdot2\cdot1\cdot1 = 6 \Rightarrow \mathrm{(E)}</math>
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==Solution 2==
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<math> 3! \cdot 5! \cdot 7! = (3\cdot2\cdot1) \cdot (5\cdot4\cdot3\cdot2\cdot1) \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)</math>
  
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===Solution 2===
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In the expression, we notice that there are 3 <math>3's</math>, 3 <math>2's</math>, and 3 <math>1's</math>. This gives us our first 3 cubes: <math>3^3</math>, <math>2^3</math>, and <math>1^3</math>.
  
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If you factor <math>3! \cdot5 ! \cdot 7!</math> You get
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However, we can also multiply smaller numbers in the expression to make bigger expressions. For example, <math>(2 \cdot 2) \cdot 4 \cdot 4=4 \cdot 4 \cdot 4 = 4^3</math> (one 2 comes from the <math>3!</math>, and the other from the <math>5!</math>). Using this method, we find:
  
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<center><math>2^7 \cdot 3^4  \cdot 5^2</math><\center>
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<math>(3 \cdot 2) \cdot (3 \cdot 2) \cdot 6 = 6^3</math>
  
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There are 3 ways for the first factor of a cube:  <math>2^0</math>, <math>2^3</math>, and <math>2^6</math>. And the second ways are:  <math>3^0</math>, and <math>3^3</math>.
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and
  
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<math>3 \cdot 2 = 6</math>
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<math>(3 \cdot 4) \cdot (3 \cdot 4) \cdot (2 \cdot 6) = 12^3</math>
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Answer : \boxed{E}
 
  
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==See Also==
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So, we have 6 cubes total: <math>1^3 ,2^3, 3^3, 4^3, 6^3,</math> and <math>12^3</math> for a total of <math>\boxed{\textbf{(E) }6}</math> cubes
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*[[2005 AMC 10A Problems]]
 
  
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*[[2005 AMC 10A Problems/Problem 14|Previous Problem]]
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==See also==
 +
{{AMC10 box|year=2005|ab=A|num-b=14|num-a=16}}
  
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*[[2005 AMC 10A Problems/Problem 16|Next Problem]]
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[[Category:Introductory Number Theory Problems]]
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[[Category:Introductory Combinatorics Problems]]  
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*[[Factorial]]
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[[Category:Introductory Number Theory Problems]]
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{{MAA Notice}}
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*[[Prime factorization]]
 
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[[Category:Introductory Combinatorics Problems]]  
 
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[[Category:Introductory Number Theory Problems]]
 

Latest revision as of 18:29, 13 December 2021

Problem

How many positive cubes divide $3! \cdot 5! \cdot 7!$ ?

$\textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 5\qquad \textbf{(E) } 6$

Solution 1

$3! \cdot 5! \cdot 7! = (3\cdot2\cdot1) \cdot (5\cdot4\cdot3\cdot2\cdot1) \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) = 2^{8}\cdot3^{4}\cdot5^{2}\cdot7^{1}$

Therefore, a perfect cube that divides $3! \cdot 5! \cdot 7!$ must be in the form $2^{a}\cdot3^{b}\cdot5^{c}\cdot7^{d}$ where $a$, $b$, $c$, and $d$ are nonnegative multiples of $3$ that are less than or equal to $8, 5, 2$ and $1,$ respectively.

So:

$a\in\{0,3,6\}$ ($3$ possibilities)

$b\in\{0,3\}$ ($2$ possibilities)

$c\in\{0\}$ ($1$ possibility)

$d\in\{0\}$($1$ possibility)

So the number of perfect cubes that divide $3! \cdot 5! \cdot 7!$ is $3\cdot2\cdot1\cdot1 = \boxed{\textbf{(E) }6}$

Solution 2

$3! \cdot 5! \cdot 7! = (3\cdot2\cdot1) \cdot (5\cdot4\cdot3\cdot2\cdot1) \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)$

In the expression, we notice that there are 3 $3's$, 3 $2's$, and 3 $1's$. This gives us our first 3 cubes: $3^3$, $2^3$, and $1^3$.

However, we can also multiply smaller numbers in the expression to make bigger expressions. For example, $(2 \cdot 2) \cdot 4 \cdot 4=4 \cdot 4 \cdot 4 = 4^3$ (one 2 comes from the $3!$, and the other from the $5!$). Using this method, we find:

$(3 \cdot 2) \cdot (3 \cdot 2) \cdot 6 = 6^3$

and

$(3 \cdot 4) \cdot (3 \cdot 4) \cdot (2 \cdot 6) = 12^3$

So, we have 6 cubes total: $1^3 ,2^3, 3^3, 4^3, 6^3,$ and $12^3$ for a total of $\boxed{\textbf{(E) }6}$ cubes

See also

2005 AMC 10A (Problems • Answer Key • Resources)
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Problem 14 		Followed by
Problem 16
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All AMC 10 Problems and Solutions

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