Difference between revisions of "2011 AMC 12B Problems/Problem 4"

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In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was <math>161.</math> What is the correct value of the product of <math>a</math> and <math>b</math>?  
 
In multiplying two positive integers <math>a</math> and <math>b</math>, Ron reversed the digits of the two-digit number <math>a</math>. His erroneous product was <math>161.</math> What is the correct value of the product of <math>a</math> and <math>b</math>?  
 
   
 
   
<math>
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<math>\textbf{(A)}\ 116 \qquad
\textbf{(A)}\ 116 \qquad
 
 
\textbf{(B)}\ 161 \qquad
 
\textbf{(B)}\ 161 \qquad
 
\textbf{(C)}\ 204 \qquad
 
\textbf{(C)}\ 204 \qquad
 
\textbf{(D)}\ 214 \qquad
 
\textbf{(D)}\ 214 \qquad
 
\textbf{(E)}\ 224 </math>
 
\textbf{(E)}\ 224 </math>
 
  
 
== Solution ==
 
== Solution ==
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<cmath> a*b = 32*7 = </cmath>
 
<cmath> a*b = 32*7 = </cmath>
  
<cmath> = \boxed{224\  \(\textbf{(E)}} </cmath>
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<cmath> = \boxed{224 \textbf{(E)}} </cmath>
  
 
== See also ==
 
== See also ==
{{AMC12 box|year=2011|before=Problem 3|num-a=5|ab=B}}
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{{AMC12 box|year=2011|num-b=3|num-a=5|ab=B}}
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{{MAA Notice}}

Latest revision as of 14:13, 19 January 2021

Problem

In multiplying two positive integers $a$ and $b$, Ron reversed the digits of the two-digit number $a$. His erroneous product was $161.$ What is the correct value of the product of $a$ and $b$?

$\textbf{(A)}\ 116 \qquad \textbf{(B)}\ 161 \qquad \textbf{(C)}\ 204 \qquad \textbf{(D)}\ 214 \qquad \textbf{(E)}\ 224$

Solution

Taking the prime factorization of $161$ reveals that it is equal to $23*7.$ Therefore, the only ways to represent $161$ as a product of two positive integers is $161*1$ and $23*7.$ Because neither $161$ nor $1$ is a two-digit number, we know that $a$ and $b$ are $23$ and $7.$ Because $23$ is a two-digit number, we know that a, with its two digits reversed, gives $23.$ Therefore, $a = 32$ and $b = 7.$ Multiplying our two correct values of $a$ and $b$ yields

\[a*b = 32*7 =\]

\[= \boxed{224 \textbf{(E)}}\]

See also

2011 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 3
Followed by
Problem 5
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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