Difference between revisions of "2018 AMC 10A Problems/Problem 19"

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==Problem==
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A number <math>m</math> is randomly selected from the set <math>\{11,13,15,17,19\}</math>, and a number <math>n</math> is randomly selected from <math>\{1999,2000,2001,\ldots,2018\}</math>. What is the probability that <math>m^n</math> has a units digit of <math>1</math>?
 
A number <math>m</math> is randomly selected from the set <math>\{11,13,15,17,19\}</math>, and a number <math>n</math> is randomly selected from <math>\{1999,2000,2001,\ldots,2018\}</math>. What is the probability that <math>m^n</math> has a units digit of <math>1</math>?
  
 
<math>\textbf{(A) }  \frac{1}{5}  \qquad        \textbf{(B) }  \frac{1}{4}  \qquad    \textbf{(C) }  \frac{3}{10}  \qquad  \textbf{(D) } \frac{7}{20} \qquad  \textbf{(E) }  \frac{2}{5} </math>
 
<math>\textbf{(A) }  \frac{1}{5}  \qquad        \textbf{(B) }  \frac{1}{4}  \qquad    \textbf{(C) }  \frac{3}{10}  \qquad  \textbf{(D) } \frac{7}{20} \qquad  \textbf{(E) }  \frac{2}{5} </math>
  
==Solution==
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== Solution 1 ==
Since we only care about the unit digit, our set <math>\{11,13,15,17,19 \}</math> can be turned into <math>\{1,3,5,7,9 \}</math>. Call this set <math>A</math> and call <math>\{1999, 2000, 2001, \cdots , 2018 \}</math> set <math>B</math>. Let's do casework on the element of <math>A</math> that we choose. Since <math>1*1=1</math>, any number from <math>B</math> can be paired with <math>1</math> to make <math>1^n</math> have a units digit of <math>1</math>. Therefore, the probability of this case happening is <math>\frac{1}{5}</math> since there is a <math>\frac{1}{5}</math> chance that the number <math>1</math> is selected from <math>A</math>. Let us consider the case where the number <math>3</math> is selected from <math>A</math>. Let's look at the unit digit when we repeatedly multiply the number <math>3</math> by itself:
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Since we only care about the units digit, our set <math>\{11,13,15,17,19 \}</math> can be turned into <math>\{1,3,5,7,9 \}</math>. Call this set <math>A</math> and call <math>\{1999, 2000, 2001, \cdots , 2018 \}</math> set <math>B</math>. Let's do casework on the element of <math>A</math> that we choose. Since <math>1\cdot 1=1</math>, any number from <math>B</math> can be paired with <math>1</math> to make <math>1^n</math> have a units digit of <math>1</math>. Therefore, the probability of this case happening is <math>\frac{1}{5}</math> since there is a <math>\frac{1}{5}</math> chance that the number <math>1</math> is selected from <math>A</math>. Let us consider the case where the number <math>3</math> is selected from <math>A</math>. Let's look at the unit digit when we repeatedly multiply the number <math>3</math> by itself:
<cmath>3*3=9</cmath>
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<cmath>3\cdot 3=9</cmath>
<cmath>9*3=7</cmath>  
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<cmath>9\cdot 3=7</cmath>  
<cmath>7*3=1</cmath>
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<cmath>7\cdot 3=1</cmath>
<cmath>1*3=3</cmath>
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<cmath>1\cdot 3=3</cmath>
 
We see that the unit digit of <math>3^x</math>, for some integer <math>x</math>, will only be <math>1</math> when <math>x</math> is a multiple of <math>4</math>. Now, let's count how many numbers in <math>B</math> are divisible by <math>4</math>. This can be done by simply listing:
 
We see that the unit digit of <math>3^x</math>, for some integer <math>x</math>, will only be <math>1</math> when <math>x</math> is a multiple of <math>4</math>. Now, let's count how many numbers in <math>B</math> are divisible by <math>4</math>. This can be done by simply listing:
 
<cmath>2000,2004,2008,2012,2016.</cmath>
 
<cmath>2000,2004,2008,2012,2016.</cmath>
There are <math>5</math> numbers in <math>B</math> divisible by <math>4</math> out of the <math>2018-1999+1=20</math> total numbers. Therefore, the probability that <math>3</math> is picked from <math>A</math> and a number divisible by <math>4</math> is picked from <math>B</math> is <math>\frac{1}{5}*\frac{5}{20}=\frac{1}{20}</math>.
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There are <math>5</math> numbers in <math>B</math> divisible by <math>4</math> out of the <math>2018-1999+1=20</math> total numbers. Therefore, the probability that <math>3</math> is picked from <math>A</math> and a number divisible by <math>4</math> is picked from <math>B</math> is <math>\frac{1}{5}\cdot \frac{5}{20}=\frac{1}{20}.</math>
 
Similarly, we can look at the repeating units digit for <math>7</math>:
 
Similarly, we can look at the repeating units digit for <math>7</math>:
<cmath>7*7=9</cmath>
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<cmath>7\cdot 7=9</cmath>
<cmath>9*7=3</cmath>
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<cmath>9\cdot 7=3</cmath>
<cmath>3*7=1</cmath>
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<cmath>3\cdot 7=1</cmath>
<cmath>1*7=7</cmath>
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<cmath>1\cdot 7=7</cmath>
 
We see that the unit digit of <math>7^y</math>, for some integer <math>y</math>, will only be <math>1</math> when <math>y</math> is a multiple of <math>4</math>. This is exactly the same conditions as our last case with <math>3</math> so the probability of this case is also <math>\frac{1}{20}</math>.  
 
We see that the unit digit of <math>7^y</math>, for some integer <math>y</math>, will only be <math>1</math> when <math>y</math> is a multiple of <math>4</math>. This is exactly the same conditions as our last case with <math>3</math> so the probability of this case is also <math>\frac{1}{20}</math>.  
Since <math>5*5=25</math> and <math>25</math> ends in <math>5</math>, the units digit of <math>5^w</math>, for some integer, <math>w</math> will always be <math>5</math>. Thus, the probability in this case is <math>0</math>.
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Since <math>5\cdot 5=25</math> and <math>25</math> ends in <math>5</math>, the units digit of <math>5^w</math>, for some integer, <math>w</math> will always be <math>5</math>. Thus, the probability in this case is <math>0</math>.
The last case we need to consider is when the number <math>9</math> is chosen from <math>A</math>. This happens with probability <math>\frac{1}{5}</math>. We list out the repeading units digit for <math>9</math> as we have done for <math>3</math> and <math>7</math>:
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The last case we need to consider is when the number <math>9</math> is chosen from <math>A</math>. This happens with probability <math>\frac{1}{5}.</math> We list out the repeating units digit for <math>9</math> as we have done for <math>3</math> and <math>7</math>:
<cmath>9*9=1</cmath>
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<cmath>9\cdot 9=1</cmath>
<cmath>1*9=9</cmath>
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<cmath>1\cdot 9=9</cmath>
We see that the units digit of <math>9^z</math>, for some integer <math>z</math>, is <math>1</math> only when <math>z</math> is an even number. From the <math>20</math> numbers in <math>B</math>, we see that exactly half of them are even. The probability in this case is <math>\frac{1}{5}*\frac{1}{2}=\frac{1}{10}.</math>
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We see that the units digit of <math>9^z</math>, for some integer <math>z</math>, is <math>1</math> only when <math>z</math> is an even number. From the <math>20</math> numbers in <math>B</math>, we see that exactly half of them are even. The probability in this case is <math>\frac{1}{5}\cdot \frac{1}{2}=\frac{1}{10}.</math>
Finally, we can ad all of our probabilities together to get  
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Finally, we can add all of our probabilities together to get  
 
<cmath>\frac{1}{5}+\frac{1}{20}+\frac{1}{20}+\frac{1}{10}=\boxed{\frac{2}{5}}.</cmath>
 
<cmath>\frac{1}{5}+\frac{1}{20}+\frac{1}{20}+\frac{1}{10}=\boxed{\frac{2}{5}}.</cmath>
  
 
~Nivek
 
~Nivek
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~very minor edits by virjoy2001
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== Solution 2 ==
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Since only the units digit is relevant, we can turn the first set into <math>\{1,3,5,7,9\}</math>. Note that <math>x^4 \equiv 1 \mod 10</math> for all odd digits <math>x</math>, except for 5. Looking at the second set, we see that it is a set of all integers between 1999 and 2018. There are 20 members of this set, which means that, <math>\mod 4</math>, this set has 5 values which correspond to <math>\{0,1,2,3\}</math>, making the probability equal for all of them. Next, check the values for which it is equal to <math>1 \mod 10</math>. There are <math>4+1+0+1+2=8</math> values for which it is equal to 1, remembering that <math>5^{4n} \equiv 1 \mod 10</math> only if <math>n=0</math>, which it is not. There are 20 values in total, and simplifying <math>\frac{8}{20}</math> gives us <math>\boxed{\frac{2}{5}}</math> or <math>\boxed{E}</math>.
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<math>QED\blacksquare</math>
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==Solution 3==
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By Euler's Theorem, we have that <math>a^{4}=1 \pmod {10}</math>, iff <math>\gcd(a,10)=1</math>.
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Hence <math>m=11,13,17,19</math>, <math>n=2000,2004,2008,2012,2016</math> work.
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Also note that <math>11^{\text{any positive integer}}\equiv 1 \pmod {10}</math> because <math>11^b=(10+1)^b=10^b+10^{b-1}1+...+10(1)+1</math>, and the latter <math>\pmod {10}</math> is clearly <math>1</math>. So <math>m=11</math>, <math>n=1999,2001,2002,2003,2005,...,2018</math> work (not counting multiples of 4 as we would be double counting if we did).
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We can also note that <math>19^{2a}\equiv 1 \pmod {10}</math> because <math>19^{2a}=361^{a}</math>, and by the same logic as why <math>11^{\text{any positive integer}}\equiv 1 \pmod {10}</math>, we are done. Hence <math>m=19</math>, and <math>n=2002, 2006, 2010, 2014, 2018</math> work (not counting any of the aforementioned cases as that would be double counting).
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We cannot make any more observations that add more <math>m^n</math> with units digit <math>1</math>, hence the number of <math>m^n</math> that have units digit one is <math>4\cdot 5+1\cdot 15+1\cdot 5=40</math>. And the total number of combinations of an element of the set of all <math>m</math> and an element of the set of all <math>n</math> is <math>5\cdot 20=100</math>. Hence the desired probability is <math>\frac{40}{100}=\frac{2}{5}</math>, which is answer choice <math>\textbf{(E)}</math>.
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~vsamc
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==Video Solution==
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https://youtu.be/M22S82Am2zM?t=630
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~IceMatrix
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==See Also==
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{{AMC10 box|year=2018|ab=A|num-b=18|num-a=20}}
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{{MAA Notice}}
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[[Category:Introductory Number Theory Problems]]
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[[Category:Introductory Probability Problems]]

Latest revision as of 17:59, 7 January 2021

Problem

A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$?

$\textbf{(A) }   \frac{1}{5}   \qquad        \textbf{(B) }   \frac{1}{4}   \qquad    \textbf{(C) }   \frac{3}{10}   \qquad   \textbf{(D) } \frac{7}{20} \qquad  \textbf{(E) }   \frac{2}{5}$

Solution 1

Since we only care about the units digit, our set $\{11,13,15,17,19 \}$ can be turned into $\{1,3,5,7,9 \}$. Call this set $A$ and call $\{1999, 2000, 2001, \cdots , 2018 \}$ set $B$. Let's do casework on the element of $A$ that we choose. Since $1\cdot 1=1$, any number from $B$ can be paired with $1$ to make $1^n$ have a units digit of $1$. Therefore, the probability of this case happening is $\frac{1}{5}$ since there is a $\frac{1}{5}$ chance that the number $1$ is selected from $A$. Let us consider the case where the number $3$ is selected from $A$. Let's look at the unit digit when we repeatedly multiply the number $3$ by itself: \[3\cdot 3=9\] \[9\cdot 3=7\] \[7\cdot 3=1\] \[1\cdot 3=3\] We see that the unit digit of $3^x$, for some integer $x$, will only be $1$ when $x$ is a multiple of $4$. Now, let's count how many numbers in $B$ are divisible by $4$. This can be done by simply listing: \[2000,2004,2008,2012,2016.\] There are $5$ numbers in $B$ divisible by $4$ out of the $2018-1999+1=20$ total numbers. Therefore, the probability that $3$ is picked from $A$ and a number divisible by $4$ is picked from $B$ is $\frac{1}{5}\cdot \frac{5}{20}=\frac{1}{20}.$ Similarly, we can look at the repeating units digit for $7$: \[7\cdot 7=9\] \[9\cdot 7=3\] \[3\cdot 7=1\] \[1\cdot 7=7\] We see that the unit digit of $7^y$, for some integer $y$, will only be $1$ when $y$ is a multiple of $4$. This is exactly the same conditions as our last case with $3$ so the probability of this case is also $\frac{1}{20}$. Since $5\cdot 5=25$ and $25$ ends in $5$, the units digit of $5^w$, for some integer, $w$ will always be $5$. Thus, the probability in this case is $0$. The last case we need to consider is when the number $9$ is chosen from $A$. This happens with probability $\frac{1}{5}.$ We list out the repeating units digit for $9$ as we have done for $3$ and $7$: \[9\cdot 9=1\] \[1\cdot 9=9\] We see that the units digit of $9^z$, for some integer $z$, is $1$ only when $z$ is an even number. From the $20$ numbers in $B$, we see that exactly half of them are even. The probability in this case is $\frac{1}{5}\cdot \frac{1}{2}=\frac{1}{10}.$ Finally, we can add all of our probabilities together to get \[\frac{1}{5}+\frac{1}{20}+\frac{1}{20}+\frac{1}{10}=\boxed{\frac{2}{5}}.\]

~Nivek

~very minor edits by virjoy2001

Solution 2

Since only the units digit is relevant, we can turn the first set into $\{1,3,5,7,9\}$. Note that $x^4 \equiv 1 \mod 10$ for all odd digits $x$, except for 5. Looking at the second set, we see that it is a set of all integers between 1999 and 2018. There are 20 members of this set, which means that, $\mod 4$, this set has 5 values which correspond to $\{0,1,2,3\}$, making the probability equal for all of them. Next, check the values for which it is equal to $1 \mod 10$. There are $4+1+0+1+2=8$ values for which it is equal to 1, remembering that $5^{4n} \equiv 1 \mod 10$ only if $n=0$, which it is not. There are 20 values in total, and simplifying $\frac{8}{20}$ gives us $\boxed{\frac{2}{5}}$ or $\boxed{E}$.

$QED\blacksquare$

Solution 3

By Euler's Theorem, we have that $a^{4}=1 \pmod {10}$, iff $\gcd(a,10)=1$. Hence $m=11,13,17,19$, $n=2000,2004,2008,2012,2016$ work.

Also note that $11^{\text{any positive integer}}\equiv 1 \pmod {10}$ because $11^b=(10+1)^b=10^b+10^{b-1}1+...+10(1)+1$, and the latter $\pmod {10}$ is clearly $1$. So $m=11$, $n=1999,2001,2002,2003,2005,...,2018$ work (not counting multiples of 4 as we would be double counting if we did).

We can also note that $19^{2a}\equiv 1 \pmod {10}$ because $19^{2a}=361^{a}$, and by the same logic as why $11^{\text{any positive integer}}\equiv 1 \pmod {10}$, we are done. Hence $m=19$, and $n=2002, 2006, 2010, 2014, 2018$ work (not counting any of the aforementioned cases as that would be double counting).

We cannot make any more observations that add more $m^n$ with units digit $1$, hence the number of $m^n$ that have units digit one is $4\cdot 5+1\cdot 15+1\cdot 5=40$. And the total number of combinations of an element of the set of all $m$ and an element of the set of all $n$ is $5\cdot 20=100$. Hence the desired probability is $\frac{40}{100}=\frac{2}{5}$, which is answer choice $\textbf{(E)}$. ~vsamc

Video Solution

https://youtu.be/M22S82Am2zM?t=630 ~IceMatrix

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
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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