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Difference between revisions of "2016 AMC 10A Problems/Problem 25"

(→‎Solution 3 (Less Casework!))
m (→‎Solution 1)
 
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==Problem==
 
==Problem==
−
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>?
+
How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900</math>?
  
 
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math>
 
<math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math>
  
−
== Solution ==
+
==Solution 1==
  
−
===Solution 1===
+
We prime factorize <math>72,600,</math> and <math>900</math>. The prime factorizations are <math>2^3\times 3^2</math>, <math>2^3\times 3\times 5^2</math> and <math>2^2\times 3^2\times 5^2</math>, respectively. Let <math>x=2^a\times 3^b\times 5^c</math>, <math>y=2^d\times 3^e\times 5^f</math> and <math>z=2^g\times 3^h\times 5^i</math>. We know that <cmath>\max(a,d)=3</cmath> <cmath>\max(b,e)=2</cmath> <cmath>\max(a,g)=3</cmath> <cmath>\max(b,h)=1</cmath> <cmath>\max(c,i)=2</cmath> <cmath>\max(d,g)=2</cmath> <cmath>\max(e,h)=2</cmath> and <math>c=f=0</math> since <math>\text{lcm}(x,y)</math> isn't a multiple of 5. Since <math>\max(d,g)=2</math> we know that <math>a=3</math>. We also know that since <math>\max(b,h)=1</math> that <math>e=2</math>. So now some equations have become useless to us...let's take them out. <cmath>\max(b,h)=1</cmath> <cmath>\max(d,g)=2</cmath> are the only two important ones left. We do casework on each now. If <math>\max(b,h)=1</math> then <math>(b,h)=(1,0),(0,1)</math> or <math>(1,1)</math>. Similarly if <math>\max(d,g)=2</math> then <math>(d,g)=(2,0),(2,1),(2,2),(1,2),(0,2)</math>. Thus our answer is <math>5\times 3=\boxed{\textbf{(A) }15}.</math>
  
−
We prime factorize <math>72,600,</math> and <math>900</math>. The prime factorizations are <math>2^3\times 3^2</math>, <math>2^3\times 3\times 5^2</math> and <math>2^2\times 3^2\times 5^2</math>, respectively. Let <math>x=2^a\times 3^b\times 5^c</math>, <math>y=2^d\times 3^e\times 5^f</math> and <math>z=2^g\times 3^h\times 5^i</math>. We know that <cmath>\max(a,d)=3</cmath> <cmath>\max(b,e)=2</cmath> <cmath>\max(a,g)=3</cmath> <cmath>\max(b,h)=1</cmath> <cmath>\max(c,i)=2</cmath> <cmath>\max(d,g)=2</cmath> <cmath>\max(e,h)=2</cmath> and <math>c=f=0</math> since <math>\text{lcm}(x,y)</math> isn't a multiple of 5. Since <math>\max(d,g)=2</math> we know that <math>a=3</math>. We also know that since <math>\max(b,h)=1</math> that <math>e=2</math>. So now some equations have become useless to us...let's take them out. <cmath>\max(b,h)=1</cmath> <cmath>\max(d,g)=2</cmath> are the only two important ones left. We do casework on each now. If <math>\max(b,h)=1</math> then <math>(b,h)=(1,0),(0,1)</math> or <math>(1,1)</math>. Similarly if <math>\max(d,g)=2</math> then <math>(d,g)=(2,0),(2,1),(2,2),(1,2),(0,2)</math>. Thus our answer is <math>5\times 3=\boxed{15 \text{(A)}}</math>.
+
==Solution 2==
−
 
 
−
===Solution 2===
 
  
 
It is well known that if the <math>\text{lcm}(a,b)=c</math> and <math>c</math> can be written as <math>p_1^ap_2^bp_3^c\dots</math>, then the highest power of all prime numbers <math>p_1,p_2,p_3\dots</math> must divide into either <math>a</math> and/or <math>b</math>. Or else a lower <math>c_0=p_1^{a-\epsilon}p_2^{b-\epsilon}p_3^{c-\epsilon}\dots</math> is the <math>\text{lcm}</math>.
 
It is well known that if the <math>\text{lcm}(a,b)=c</math> and <math>c</math> can be written as <math>p_1^ap_2^bp_3^c\dots</math>, then the highest power of all prime numbers <math>p_1,p_2,p_3\dots</math> must divide into either <math>a</math> and/or <math>b</math>. Or else a lower <math>c_0=p_1^{a-\epsilon}p_2^{b-\epsilon}p_3^{c-\epsilon}\dots</math> is the <math>\text{lcm}</math>.
Line 17: Line 15:
 
So <math>x=8,24</math>.
 
So <math>x=8,24</math>.
  
−
<math>y</math> can be <math>9,18,36</math> in both cases of <math>x</math> but NOT <math>72</math> because <math>\text{lcm}{y,z}=900</math> and <math>72\nmid 900</math>.
+
<math>y</math> can be <math>9,18,36</math> in both cases of <math>x</math> but NOT <math>72</math> because <math>\text{lcm}(y,z)=900</math> and <math>72\nmid 900</math>.
  
 
So there are six sets of <math>x,y</math> and we will list all possible values of <math>z</math> based on those.
 
So there are six sets of <math>x,y</math> and we will list all possible values of <math>z</math> based on those.
Line 36: Line 34:
 
Counting the cases, <math>1+1+3+2+2+6=\boxed{\textbf{(A) }15}.</math>
 
Counting the cases, <math>1+1+3+2+2+6=\boxed{\textbf{(A) }15}.</math>
  
−
===Solution 3 (Less Casework!)===
+
==Solution 3 (Less Casework!)==
 
As said in previous solutions, start by factoring <math>72, 600,</math> and <math>900</math>. The prime factorizations are as follows: <cmath>72=2^3\cdot 3^2,</cmath> <cmath>600=2^3\cdot 3\cdot 5^2,</cmath> <cmath> \text{and } 900=2^2\cdot 3^2\cdot 5^2</cmath>
 
As said in previous solutions, start by factoring <math>72, 600,</math> and <math>900</math>. The prime factorizations are as follows: <cmath>72=2^3\cdot 3^2,</cmath> <cmath>600=2^3\cdot 3\cdot 5^2,</cmath> <cmath> \text{and } 900=2^2\cdot 3^2\cdot 5^2</cmath>
 +
To organize <math>x,y, \text{ and } z</math> and their respective LCMs in a simpler way, we can draw a triangle as follows such that <math>x,y, \text{and } z</math> are the vertices and the LCMs are on the edges.
 +
 +
<asy>
 +
//Variable Declarations
 +
defaultpen(0.45);
 +
size(200pt);
 +
fontsize(15pt);
 +
pair X, Y, Z;
 +
real R;
 +
path tri;
 +
 +
//Variable Definitions
 +
R = 1;
 +
X = R*dir(90);
 +
Y = R*dir(210);
 +
Z = R*dir(-30);
 +
tri = X--Y--Z--cycle;
 +
 +
//Diagram
 +
draw(tri);
 +
label("$x$",X,N);
 +
label("$y$",Y,SW);
 +
label("$z$",Z,SE);
 +
label("$2^33^25^0$",X--Y,2W);
 +
label("$2^33^15^2$",X--Z,2E);
 +
label("$2^23^25^2$",Y--Z,2S);
 +
</asy>
 +
 +
Now we can split this triangle into three separate ones for each of the three different prime factors <math>2,3, \text{and } 5</math>.
  
−
To organize <math>x,y, \text{ and } z</math> and their respective LCMs in a simpler way, we can draw a triangle as follows such that <math>x,y, \text{and } z</math> are the vertices and the LCMs are on the edges.
+
<asy>
−
<cmath>[\textbf{\emph{insert diagram here}}]</cmath>
+
//Variable Declarations
 +
defaultpen(0.45);
 +
size(200pt);
 +
fontsize(15pt);
 +
pair X, Y, Z;
 +
real R;
 +
path tri;
 +
 
 +
//Variable Definitions
 +
R = 1;
 +
X = R*dir(90);
 +
Y = R*dir(210);
 +
Z = R*dir(-30);
 +
tri = X--Y--Z--cycle;
 +
 
 +
//Diagram
 +
draw(tri);
 +
label("$x$",X,N);
 +
label("$y$",Y,SW);
 +
label("$z$",Z,SE);
 +
label("$2^3$",X--Y,2W);
 +
label("$2^3$",X--Z,2E);
 +
label("$2^2$",Y--Z,2S);
 +
</asy>
 +
 
 +
Analyzing for powers of <math>2</math>, it is quite obvious that <math>x</math> must have <math>2^3</math> as one of its factors since neither <math>y \text{ nor } z</math> can have a power of <math>2</math> exceeding <math>2</math>. Turning towards the vertices <math>y</math> and <math>z</math>, we know at least one of them must have <math>2^2</math> as its factors. Therefore, we have <math>5</math> ways for the powers of <math>2</math> for <math>y \text{ and } z</math> since the only ones that satisfy the previous conditions are for ordered pairs <math>(y,z) \{(2,0)(2,1)(0,2)(1,2)(2,2)\}</math>.
 +
 
 +
<asy>
 +
//Variable Declarations
 +
defaultpen(0.45);
 +
size(200pt);
 +
fontsize(15pt);
 +
pair X, Y, Z;
 +
real R;
 +
path tri;
  
−
Now we can split this triangle into three separate ones for each of the three different prime factors <math>2,3, \text{and} 5</math>.
+
//Variable Definitions
−
<cmath>[\textbf{\emph{insert diagram here for powers of 2}}]</cmath>
+
R = 1;
 +
X = R*dir(90);
 +
Y = R*dir(210);
 +
Z = R*dir(-30);
 +
tri = X--Y--Z--cycle;
  
−
Analyzing for powers of <math>2</math>, is is quite obvious that <math>x</math> must have <math>2^3</math> as one of it's factors, since neither <math>y \text{ nor } z</math> can have a power of <math>2</math> exceeding <math>2</math>. Turning towards the vertices <math>y \text{and} z</math>, we know at least one of them must have <math>2^2</math> as its factors. Therefore, we have <math>5</math> ways for the powers of <math>2</math> for <math>y \text{ and } z</math> since the only ones that satisfy the previous conditions are for ordered pairs <math>(y,z) \{(2,0)(2,1)(0,2)(1,2)(2,2)\}</math>.
+
//Diagram
 +
draw(tri);
 +
label("$x$",X,N);
 +
label("$y$",Y,SW);
 +
label("$z$",Z,SE);
 +
label("$3^2$",X--Y,2W);
 +
label("$3^1$",X--Z,2E);
 +
label("$3^2$",Y--Z,2S);
 +
</asy>
  
−
Now we can move on to powers of <math>3</math>.
+
Using the same logic as we did for powers of <math>2</math>, it becomes quite easy to note that <math>y</math> must have <math>3^2</math> as one of its factors. Moving onto <math>x \text{ and } z</math>, we can use the same logic to find the only ordered pairs <math>(x,z)</math> that will work are <math>\{(1,0)(0,1)(1,1)\}</math>.
−
<cmath>[\textbf{\emph{insert diagram here for powers of 3}}]</cmath>
 
−
Using the same logic as we did for powers of <math>2</math>, is becomes quite easy to note that <math>y</math> must have <math>3^2</math> as one of it's factors. Moving onto <math>x \text{ and } z</math>, we can use the same logic to find the only ordered pairs <math>(x,z)</math> that will work are <math>\{(1,0)(0,1)(1,1)\}</math>.  
 
  
 
The final and last case is the powers of <math>5</math>.
 
The final and last case is the powers of <math>5</math>.
−
<cmath>[\textbf{\emph{insert diagram here for powers of 5}}]</cmath>
+
 
−
This is actually quite a simple case, since we know <math>z</math> must have <math>5^2</math> as part of its factorization while <math>x \text{ and } y</math> cannot have a factor of <math>5</math> in their prime factorization.
+
<asy>
 +
//Variable Declarations
 +
defaultpen(0.45);
 +
size(200pt);
 +
fontsize(15pt);
 +
pair X, Y, Z;
 +
real R;
 +
path tri;
 +
 
 +
//Variable Definitions
 +
R = 1;
 +
X = R*dir(90);
 +
Y = R*dir(210);
 +
Z = R*dir(-30);
 +
tri = X--Y--Z--cycle;
 +
 
 +
//Diagram
 +
draw(tri);
 +
label("$x$",X,N);
 +
label("$y$",Y,SW);
 +
label("$z$",Z,SE);
 +
label("$5^0$",X--Y,2W);
 +
label("$5^2$",X--Z,2E);
 +
label("$5^2$",Y--Z,2S);
 +
</asy>
 +
 
 +
This is actually quite a simple case since we know <math>z</math> must have <math>5^2</math> as part of its factorization while <math>x \text{ and } y</math> cannot have a factor of <math>5</math> in their prime factorization.
  
 
Multiplying all the possible arrangements for prime factors <math>2,3, \text{ and } 5</math>, we get the answer:
 
Multiplying all the possible arrangements for prime factors <math>2,3, \text{ and } 5</math>, we get the answer:
−
<cmath>5\cdot3\cdot1=\boxed{\textbf{(A) }15}</cmath>.
+
<cmath>5\cdot3\cdot1=\boxed{\textbf{(A) }15}</cmath>
 +
 
 +
(Diagrams by ColtsFan10)
 +
 
 +
==Video Solution==
 +
 
 +
https://www.youtube.com/watch?v=ja1KZ8tVwI8
  
 
==See Also==
 
==See Also==
 +
 
{{AMC10 box|year=2016|ab=A|num-b=24|after=Last Problem}}
 
{{AMC10 box|year=2016|ab=A|num-b=24|after=Last Problem}}
 
{{AMC12 box|year=2016|ab=A|num-b=21|num-a=23}}
 
{{AMC12 box|year=2016|ab=A|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:47, 16 October 2024

Problem

How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$?

$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64$

Solution 1

We prime factorize $72,600,$ and $900$. The prime factorizations are $2^3\times 3^2$, $2^3\times 3\times 5^2$ and $2^2\times 3^2\times 5^2$, respectively. Let $x=2^a\times 3^b\times 5^c$, $y=2^d\times 3^e\times 5^f$ and $z=2^g\times 3^h\times 5^i$. We know that \[\max(a,d)=3\] \[\max(b,e)=2\] \[\max(a,g)=3\] \[\max(b,h)=1\] \[\max(c,i)=2\] \[\max(d,g)=2\] \[\max(e,h)=2\] and $c=f=0$ since $\text{lcm}(x,y)$ isn't a multiple of 5. Since $\max(d,g)=2$ we know that $a=3$. We also know that since $\max(b,h)=1$ that $e=2$. So now some equations have become useless to us...let's take them out. \[\max(b,h)=1\] \[\max(d,g)=2\] are the only two important ones left. We do casework on each now. If $\max(b,h)=1$ then $(b,h)=(1,0),(0,1)$ or $(1,1)$. Similarly if $\max(d,g)=2$ then $(d,g)=(2,0),(2,1),(2,2),(1,2),(0,2)$. Thus our answer is $5\times 3=\boxed{\textbf{(A) }15}.$

Solution 2

It is well known that if the $\text{lcm}(a,b)=c$ and $c$ can be written as $p_1^ap_2^bp_3^c\dots$, then the highest power of all prime numbers $p_1,p_2,p_3\dots$ must divide into either $a$ and/or $b$. Or else a lower $c_0=p_1^{a-\epsilon}p_2^{b-\epsilon}p_3^{c-\epsilon}\dots$ is the $\text{lcm}$.

Start from $x$:$\text{lcm}(x,y)=72$ so $8\mid x$ or $9\mid x$ or both. But $9\nmid x$ because $\text{lcm}(x,z)=600$ and $9\nmid 600$. So $x=8,24$.

$y$ can be $9,18,36$ in both cases of $x$ but NOT $72$ because $\text{lcm}(y,z)=900$ and $72\nmid 900$.

So there are six sets of $x,y$ and we will list all possible values of $z$ based on those.

$25\mid z$ because $z$ must source all powers of $5$. $z\in\{25,50,75,100,150,300\}$. $z\ne\{200,225\}$ because of $\text{lcm}$ restrictions.

By different sourcing of powers of $2$ and $3$,

\[(8,9):z=300\] \[(8,18):z=300\] \[(8,36):z=75,150,300\] \[(24,9):z=100,300\] \[(24,18):z=100,300\] \[(24,36):z=25,50,75,100,150,300\]

$z=100$ is "enabled" by $x$ sourcing the power of $3$. $z=75,150$ is uncovered by $y$ sourcing all powers of $2$. And $z=25,50$ is uncovered by $x$ and $y$ both at full power capacity.

Counting the cases, $1+1+3+2+2+6=\boxed{\textbf{(A) }15}.$

Solution 3 (Less Casework!)

As said in previous solutions, start by factoring $72, 600,$ and $900$. The prime factorizations are as follows: \[72=2^3\cdot 3^2,\] \[600=2^3\cdot 3\cdot 5^2,\] \[\text{and } 900=2^2\cdot 3^2\cdot 5^2\] To organize $x,y, \text{ and } z$ and their respective LCMs in a simpler way, we can draw a triangle as follows such that $x,y, \text{and } z$ are the vertices and the LCMs are on the edges.

[asy] //Variable Declarations defaultpen(0.45); size(200pt); fontsize(15pt); pair X, Y, Z; real R; path tri; //Variable Definitions R = 1; X = R*dir(90); Y = R*dir(210); Z = R*dir(-30); tri = X--Y--Z--cycle; //Diagram draw(tri); label("$x$",X,N); label("$y$",Y,SW); label("$z$",Z,SE); label("$2^33^25^0$",X--Y,2W); label("$2^33^15^2$",X--Z,2E); label("$2^23^25^2$",Y--Z,2S); [/asy]

Now we can split this triangle into three separate ones for each of the three different prime factors $2,3, \text{and } 5$.

[asy] //Variable Declarations defaultpen(0.45); size(200pt); fontsize(15pt); pair X, Y, Z; real R; path tri; //Variable Definitions R = 1; X = R*dir(90); Y = R*dir(210); Z = R*dir(-30); tri = X--Y--Z--cycle; //Diagram draw(tri); label("$x$",X,N); label("$y$",Y,SW); label("$z$",Z,SE); label("$2^3$",X--Y,2W); label("$2^3$",X--Z,2E); label("$2^2$",Y--Z,2S); [/asy]

Analyzing for powers of $2$, it is quite obvious that $x$ must have $2^3$ as one of its factors since neither $y \text{ nor } z$ can have a power of $2$ exceeding $2$. Turning towards the vertices $y$ and $z$, we know at least one of them must have $2^2$ as its factors. Therefore, we have $5$ ways for the powers of $2$ for $y \text{ and } z$ since the only ones that satisfy the previous conditions are for ordered pairs $(y,z) \{(2,0)(2,1)(0,2)(1,2)(2,2)\}$.

[asy] //Variable Declarations defaultpen(0.45); size(200pt); fontsize(15pt); pair X, Y, Z; real R; path tri; //Variable Definitions R = 1; X = R*dir(90); Y = R*dir(210); Z = R*dir(-30); tri = X--Y--Z--cycle; //Diagram draw(tri); label("$x$",X,N); label("$y$",Y,SW); label("$z$",Z,SE); label("$3^2$",X--Y,2W); label("$3^1$",X--Z,2E); label("$3^2$",Y--Z,2S); [/asy]

Using the same logic as we did for powers of $2$, it becomes quite easy to note that $y$ must have $3^2$ as one of its factors. Moving onto $x \text{ and } z$, we can use the same logic to find the only ordered pairs $(x,z)$ that will work are $\{(1,0)(0,1)(1,1)\}$.

The final and last case is the powers of $5$.

[asy] //Variable Declarations defaultpen(0.45); size(200pt); fontsize(15pt); pair X, Y, Z; real R; path tri; //Variable Definitions R = 1; X = R*dir(90); Y = R*dir(210); Z = R*dir(-30); tri = X--Y--Z--cycle; //Diagram draw(tri); label("$x$",X,N); label("$y$",Y,SW); label("$z$",Z,SE); label("$5^0$",X--Y,2W); label("$5^2$",X--Z,2E); label("$5^2$",Y--Z,2S); [/asy]

This is actually quite a simple case since we know $z$ must have $5^2$ as part of its factorization while $x \text{ and } y$ cannot have a factor of $5$ in their prime factorization.

Multiplying all the possible arrangements for prime factors $2,3, \text{ and } 5$, we get the answer: \[5\cdot3\cdot1=\boxed{\textbf{(A) }15}\]

(Diagrams by ColtsFan10)

Video Solution

https://www.youtube.com/watch?v=ja1KZ8tVwI8

See Also

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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
All AMC 10 Problems and Solutions
2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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
All AMC 12 Problems and Solutions

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