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2012 AMC 10B Problems/Problem 10: Difference between revisions

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== Problem 10 ==
== Problem 10 ==
How many ordered pairs of positive integers <math>(M,N)</math> satisfy the equation <math>\frac {M}{6}</math>    =     <math>\frac{6}{N}</math>?
How many ordered pairs of positive integers <math>(M,N)</math> satisfy the equation <math>\frac{M}{6}=\frac{6}{N}?</math>


<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math>
<math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10 </math>
Line 6: Line 6:
[[2012 AMC 10B Problems/Problem 10|Solution]]
[[2012 AMC 10B Problems/Problem 10|Solution]]


== Solution ==
== Solution 1 ==
Cross-multiplying gives <math>MN=36.</math> We write <math>36</math> as a product of two positive integers:
<cmath>\begin{align*}
36 &= 1\cdot36 \\
&= 2\cdot18 \\
&= 3\cdot12 \\
&= 4\cdot9 \\
&= 6\cdot6.
\end{align*}</cmath>
The products <math>1\cdot36, 2\cdot18, 3\cdot12,</math> and <math>4\cdot9</math> each produce <math>2</math> ordered pairs <math>(M,N),</math> as we can switch the order of the factors. The product <math>6\cdot6</math> produces <math>1</math> ordered pair <math>(M,N).</math> Together, we have <math>4\cdot2+1=\boxed{\textbf{(D)}\ 9}</math> ordered pairs <math>(M,N).</math>


<math>\frac {M}{6}</math>    =    <math>\frac{6}{N}</math>
~Rguan (Solution)


is a ratio; therefore, you can cross-multiply.
~MRENTHUSIASM (Reformatting)
 
<math>MN=36</math>
 
Now you find all the factors of 36:
 
<math>1\times36=36</math>
 
<math>2\times18=36</math>
 
<math>3\times12=36</math>
 
<math>4\times9=36</math>
 
<math>6\times6=36</math>.
 
Now you can reverse the order of the factors for all of the ones listed above, because they are ordered pairs except for 6*6 since it is the same if you reverse the order.
 
<math>4\cdot 2+1=9</math>
 
<math>\boxed{\textbf{(D)}\ 9}</math>


==See Also==
==See Also==

Revision as of 19:43, 3 September 2021

Problem 10

How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}?$

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$

Solution

Solution 1

Cross-multiplying gives $MN=36.$ We write $36$ as a product of two positive integers: \begin{align*} 36 &= 1\cdot36 \\ &= 2\cdot18 \\ &= 3\cdot12 \\ &= 4\cdot9 \\ &= 6\cdot6. \end{align*} The products $1\cdot36, 2\cdot18, 3\cdot12,$ and $4\cdot9$ each produce $2$ ordered pairs $(M,N),$ as we can switch the order of the factors. The product $6\cdot6$ produces $1$ ordered pair $(M,N).$ Together, we have $4\cdot2+1=\boxed{\textbf{(D)}\ 9}$ ordered pairs $(M,N).$

~Rguan (Solution)

~MRENTHUSIASM (Reformatting)

See Also

2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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

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