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Difference between revisions of "2022 AMC 10B Problems/Problem 8"
(→‎Solution 5 (Very Fast System of Equations))
 
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&\{991,992,993,\ldots,1000\}.
 
&\{991,992,993,\ldots,1000\}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
 
 
How many of these sets contain exactly two multiples of <math>7</math>?
 
How many of these sets contain exactly two multiples of <math>7</math>?
  
 
<math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50</math>
 
<math>\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50</math>
  
−
==Solution==
+
==Solution 1 (Casework)==
−
We apply casework to this problem:
+
We apply casework to this problem. The only sets that contain two multiples of seven are those for which:
  
 
<ol style="margin-left: 1.5em;">
 
<ol style="margin-left: 1.5em;">
−
   <li>The multiples of <math>7</math> are <math>1\pmod{10}</math> and <math>8\pmod{10}.</math></li><p>
+
   <li>The multiples of <math>7</math> are <math>1\pmod{10}</math> and <math>8\pmod{10}.</math> That is, the first and eighth elements of such sets are multiples of <math>7.</math></li><p>
−
   <li>The multiples of <math>7</math> are <math>2\pmod{10}</math> and <math>9\pmod{10}.</math></li><p>
+
The first element is <math>1+10k</math> for some integer <math>0\leq k\leq99.</math> It is a multiple of <math>7</math> when <math>k=2,9,16,\ldots,93.</math>
−
   <li>The multiples of <math>7</math> are <math>3\pmod{10}</math> and <math>0\pmod{10}.</math></li><p>
+
   <li>The multiples of <math>7</math> are <math>2\pmod{10}</math> and <math>9\pmod{10}.</math> That is, the second and ninth elements of such sets are multiples of <math>7.</math></li><p>
 +
The second element is <math>2+10k</math> for some integer <math>0\leq k\leq99.</math> It is a multiple of <math>7</math> when <math>k=4,11,18,\ldots,95.</math>
 +
   <li>The multiples of <math>7</math> are <math>3\pmod{10}</math> and <math>0\pmod{10}.</math> That is, the third and tenth elements of such sets are multiples of <math>7.</math></li><p>
 +
The third element is <math>3+10k</math> for some integer <math>0\leq k\leq99.</math> It is a multiple of <math>7</math> when <math>k=6,13,20,\ldots,97.</math>
 
</ol>
 
</ol>
 
Each case has <math>\left\lfloor\frac{100}{7}\right\rfloor=14</math> sets. Therefore, the answer is <math>14\cdot3=\boxed{\textbf{(B)}\ 42}.</math>
 
Each case has <math>\left\lfloor\frac{100}{7}\right\rfloor=14</math> sets. Therefore, the answer is <math>14\cdot3=\boxed{\textbf{(B)}\ 42}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
 +
 +
==Solution 2 (Find A Pattern)==
 +
We find a pattern.
 +
<cmath>\begin{align*}
 +
&\{1,2,3,\ldots,10\}, \\
 +
&\{11,12,13,\ldots,20\},\\
 +
&\{21,22,23,\ldots,30\},\\
 +
&\vdots\\
 +
&\{991,992,993,\ldots,1000\}.
 +
\end{align*}</cmath>
 +
We can figure out that the first set has <math>1</math> multiple of <math>7</math>. The second set also has <math>1</math> multiple of <math>7</math>. The third set has <math>2</math> multiples of <math>7</math>. The fourth set has <math>1</math> multiple of <math>7</math>. The fifth set has <math>2</math> multiples of <math>7</math>. The sixth set has <math>1</math> multiple of <math>7</math>. The seventh set has <math>1</math> multiple of <math>7</math>. The eighth set has <math>2</math> multiples of <math>7</math>. Disregarding the first set and then calculating this pattern further, we can see (reasonably) that it repeats for each <math>1</math> sets.
 +
We see that the pattern for the number of multiples per <math>7</math> sets (again, disregarding the first set) goes: <math>1,2,1,2,1,1,2.</math> So, for every <math>7</math> sets after the first, there are three sets with <math>2</math> multiples of <math>7</math>. We calculate <math>\left\lfloor\frac{100-1}{7}\right\rfloor</math> and multiply that by <math>3</math>. (We also disregard the remainder of <math>1</math> since it doesn't add any extra sets with <math>2</math> multiples of <math>7</math>.). We get <math>14\cdot3= \boxed{\textbf{(B) }42}</math>.
 +
 +
~(edited by) mihikamishra
 +
 +
~(edited by) MiniGlasses2009
 +
 +
==Solution 3 (Fastest)==
 +
Each set contains exactly <math>1</math> or <math>2</math> multiples of <math>7</math>.
 +
 +
There are <math>\dfrac{1000}{10}=100</math> total sets and <math>\left\lfloor\dfrac{1000}{7}\right\rfloor = 142</math> multiples of <math>7</math>. Thus, there are <math>142-100=\boxed{\textbf{(B) }42}</math> sets with <math>2</math> multiples of <math>7</math>.
 +
 +
~BrandonZhang202415
 +
 +
== Solution 4 (Simple Counting, Similar to Solution 1) ==
 +
 +
Consecutive multiples of <math>7</math> must differ by <math>7</math>. So, if a set <math>\{\ldots1,\ldots2,\ldots3,(\ldots),\ldots8,\ldots9,\ldots0\}</math> contains two multiples of <math>7</math>, they must end with the digits <math>1</math> and <math>8</math>, <math>2</math> and <math>9</math>, or <math>3</math> and <math>0</math>. This reduces the problem to counting the amount of multiples of <math>7</math> less than <math>1000</math> that end with <math>1</math>, <math>2</math>, and <math>3</math>.
 +
 +
The first multiple of <math>7</math> that ends with <math>1</math> is <math>21</math>. The next multiple that ends with <math>1</math> occurs <math>70</math> later, since that is the smallest multiple of <math>7</math> we can add to <math>21</math> without affecting the last digit. The greatest number of <math>70</math>'s we can add to <math>21</math> while keeping it less than <math>1000</math> is <math>13</math>, because <math>21 + 70(13) = 931</math>. Therefore, the set of multiples of <math>7</math> less than <math>1000</math> ending with <math>1</math> is <math>\{21 + 70(0), 21+70(1),\ldots,21+70(13)\}</math>, meaning there are <math>14</math> of these particular multiples. We can use the same reasoning to count the multiples of <math>7</math> that end with <math>2</math> and <math>3</math>.
 +
 +
The first multiple of <math>7</math> that ends with <math>2</math> is <math>42</math>. The greatest number of <math>70</math>'s we can add to <math>42</math> here is also <math>13</math>, since <math>42 + 70(13) = 952</math>. The set of multiples of <math>7</math> less than <math>1000</math> ending with <math>2</math> is <math>\{42 + 70(0), 42+70(1),\ldots,42+70(13)\}</math>, giving <math>14</math> multiples.
 +
 +
The first multiple of <math>7</math> that ends with <math>3</math> is <math>63</math>. The greatest number of <math>70</math>'s we can add to <math>63</math> here is yet again <math>13</math>, since <math>63 + 70(13) = 973</math>. The set of multiples of <math>7</math> less than <math>1000</math> ending with <math>3</math> is <math>\{63 + 70(0), 63+70(1),\ldots,63+70(13)\}</math>, giving another <math>14</math> multiples.
 +
 +
In total, there are <math>14 + 14 + 14 = 42</math> of these multiples, and so <math>\boxed{\textbf{(B) }42}</math> sets with two multiples of <math>7</math>.
 +
 +
~marsus16112
 +
 +
== Solution 5 (Very Fast System of Equations) ==
 +
 +
Let <math>a</math> be the number of sets with <math>1</math> multiple of <math>7</math> and <math>b</math> be the number of sets with <math>2</math> multiples of <math>7</math>. Note that it is impossible for a set to have more than two multiples of <math>7</math>.
 +
 +
Since there are a total of <math>100</math> sets, <math>a+b=100</math>. Also, since there are <math>\lfloor \frac{1000}{7}\rfloor = 142</math> multiples of <math>7</math> between <math>1</math> and <math>1000</math>, we must have <math>a+2b=142</math>.
 +
 +
Solving the system of equations
 +
 +
<cmath>a+b=100</cmath>
 +
<cmath>a+2b=142</cmath>
 +
 +
for <math>b</math> gives us the answer of <math>\boxed{\textbf{(B) }42}</math>.
 +
 +
~FIREDRAGONMATH16
 +
~scrares (minor edit)
 +
 +
==Solution 6 (reasonable count)==
 +
 +
(Similar to Solution 2, but a little more intuitive and less numbers.) Note that this system loops every cycle of length <math>70</math>, or <math>7</math> such sets. From <math>1</math> to <math>70</math>, the multiples of <math>7</math> are <math>7</math>, <math>14</math>, <math>21</math>, <math>28</math>, <math>35</math>, <math>42</math>, <math>49</math>, <math>56</math>, <math>63</math>, and <math>70</math>; note that <math>(21,28),(42,49),</math> and <math>(63,70)</math> are in the same sets. Thus, for every <math>7</math> sets, we have <math>3</math> sets with exactly two multiples of <math>7</math>. We have <math>100</math> sets, which is <math>98+2=7\cdot14+2</math>; the first <math>98</math> sets contain <math>\dfrac37(7\cdot14)=42</math> desired sets. The last two sets comprise the integers from <math>981</math> to <math>999</math>; the multiples of <math>7</math> here are <math>987</math> and <math>994</math>. Neither of the two last sets contains two multiples of <math>7</math>, so our answer is simply <math>\boxed{\textbf{(D) }42}</math>.
 +
 +
~Technodoggo
 +
 +
==Video Solution (🚀Under 3 min🚀)==
 +
https://youtu.be/PdyKJ1p9Y2w
 +
 +
~Education, the Study of Everything
 +
 +
==Video Solution(1-16)==
 +
https://youtu.be/SCwQ9jUfr0g
 +
 +
~~Hayabusa1
 +
 +
==Video Solution by Interstigation==
 +
https://youtu.be/_KNR0JV5rdI?t=884
  
 
== See Also ==
 
== See Also ==

Latest revision as of 04:57, 11 November 2024

The following problem is from both the 2022 AMC 10B #8 and 2022 AMC 12B #6, so both problems redirect to this page.

Problem

Consider the following $100$ sets of $10$ elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} How many of these sets contain exactly two multiples of $7$?

$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 42\qquad\textbf{(C)}\ 43\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 50$

Solution 1 (Casework)

We apply casework to this problem. The only sets that contain two multiples of seven are those for which:

  1. The multiples of $7$ are $1\pmod{10}$ and $8\pmod{10}.$ That is, the first and eighth elements of such sets are multiples of $7.$

    2. The first element is $1+10k$ for some integer $0\leq k\leq99.$ It is a multiple of $7$ when $k=2,9,16,\ldots,93.$

  2. The multiples of $7$ are $2\pmod{10}$ and $9\pmod{10}.$ That is, the second and ninth elements of such sets are multiples of $7.$

    3. The second element is $2+10k$ for some integer $0\leq k\leq99.$ It is a multiple of $7$ when $k=4,11,18,\ldots,95.$

  3. The multiples of $7$ are $3\pmod{10}$ and $0\pmod{10}.$ That is, the third and tenth elements of such sets are multiples of $7.$

    4. The third element is $3+10k$ for some integer $0\leq k\leq99.$ It is a multiple of $7$ when $k=6,13,20,\ldots,97.$

Each case has $\left\lfloor\frac{100}{7}\right\rfloor=14$ sets. Therefore, the answer is $14\cdot3=\boxed{\textbf{(B)}\ 42}.$

~MRENTHUSIASM

Solution 2 (Find A Pattern)

We find a pattern. \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21,22,23,\ldots,30\},\\ &\vdots\\ &\{991,992,993,\ldots,1000\}. \end{align*} We can figure out that the first set has $1$ multiple of $7$. The second set also has $1$ multiple of $7$. The third set has $2$ multiples of $7$. The fourth set has $1$ multiple of $7$. The fifth set has $2$ multiples of $7$. The sixth set has $1$ multiple of $7$. The seventh set has $1$ multiple of $7$. The eighth set has $2$ multiples of $7$. Disregarding the first set and then calculating this pattern further, we can see (reasonably) that it repeats for each $1$ sets. We see that the pattern for the number of multiples per $7$ sets (again, disregarding the first set) goes: $1,2,1,2,1,1,2.$ So, for every $7$ sets after the first, there are three sets with $2$ multiples of $7$. We calculate $\left\lfloor\frac{100-1}{7}\right\rfloor$ and multiply that by $3$. (We also disregard the remainder of $1$ since it doesn't add any extra sets with $2$ multiples of $7$.). We get $14\cdot3= \boxed{\textbf{(B) }42}$.

~(edited by) mihikamishra

~(edited by) MiniGlasses2009

Solution 3 (Fastest)

Each set contains exactly $1$ or $2$ multiples of $7$.

There are $\dfrac{1000}{10}=100$ total sets and $\left\lfloor\dfrac{1000}{7}\right\rfloor = 142$ multiples of $7$. Thus, there are $142-100=\boxed{\textbf{(B) }42}$ sets with $2$ multiples of $7$.

~BrandonZhang202415

Solution 4 (Simple Counting, Similar to Solution 1)

Consecutive multiples of $7$ must differ by $7$. So, if a set $\{\ldots1,\ldots2,\ldots3,(\ldots),\ldots8,\ldots9,\ldots0\}$ contains two multiples of $7$, they must end with the digits $1$ and $8$, $2$ and $9$, or $3$ and $0$. This reduces the problem to counting the amount of multiples of $7$ less than $1000$ that end with $1$, $2$, and $3$.

The first multiple of $7$ that ends with $1$ is $21$. The next multiple that ends with $1$ occurs $70$ later, since that is the smallest multiple of $7$ we can add to $21$ without affecting the last digit. The greatest number of $70$'s we can add to $21$ while keeping it less than $1000$ is $13$, because $21 + 70(13) = 931$. Therefore, the set of multiples of $7$ less than $1000$ ending with $1$ is $\{21 + 70(0), 21+70(1),\ldots,21+70(13)\}$, meaning there are $14$ of these particular multiples. We can use the same reasoning to count the multiples of $7$ that end with $2$ and $3$.

The first multiple of $7$ that ends with $2$ is $42$. The greatest number of $70$'s we can add to $42$ here is also $13$, since $42 + 70(13) = 952$. The set of multiples of $7$ less than $1000$ ending with $2$ is $\{42 + 70(0), 42+70(1),\ldots,42+70(13)\}$, giving $14$ multiples.

The first multiple of $7$ that ends with $3$ is $63$. The greatest number of $70$'s we can add to $63$ here is yet again $13$, since $63 + 70(13) = 973$. The set of multiples of $7$ less than $1000$ ending with $3$ is $\{63 + 70(0), 63+70(1),\ldots,63+70(13)\}$, giving another $14$ multiples.

In total, there are $14 + 14 + 14 = 42$ of these multiples, and so $\boxed{\textbf{(B) }42}$ sets with two multiples of $7$.

~marsus16112

Solution 5 (Very Fast System of Equations)

Let $a$ be the number of sets with $1$ multiple of $7$ and $b$ be the number of sets with $2$ multiples of $7$. Note that it is impossible for a set to have more than two multiples of $7$.

Since there are a total of $100$ sets, $a+b=100$. Also, since there are $\lfloor \frac{1000}{7}\rfloor = 142$ multiples of $7$ between $1$ and $1000$, we must have $a+2b=142$.

Solving the system of equations

\[a+b=100\] \[a+2b=142\]

for $b$ gives us the answer of $\boxed{\textbf{(B) }42}$.

~FIREDRAGONMATH16 ~scrares (minor edit)

Solution 6 (reasonable count)

(Similar to Solution 2, but a little more intuitive and less numbers.) Note that this system loops every cycle of length $70$, or $7$ such sets. From $1$ to $70$, the multiples of $7$ are $7$, $14$, $21$, $28$, $35$, $42$, $49$, $56$, $63$, and $70$; note that $(21,28),(42,49),$ and $(63,70)$ are in the same sets. Thus, for every $7$ sets, we have $3$ sets with exactly two multiples of $7$. We have $100$ sets, which is $98+2=7\cdot14+2$; the first $98$ sets contain $\dfrac37(7\cdot14)=42$ desired sets. The last two sets comprise the integers from $981$ to $999$; the multiples of $7$ here are $987$ and $994$. Neither of the two last sets contains two multiples of $7$, so our answer is simply $\boxed{\textbf{(D) }42}$.

~Technodoggo

Video Solution (🚀Under 3 min🚀)

https://youtu.be/PdyKJ1p9Y2w

~Education, the Study of Everything

Video Solution(1-16)

https://youtu.be/SCwQ9jUfr0g

~~Hayabusa1

Video Solution by Interstigation

https://youtu.be/_KNR0JV5rdI?t=884

See Also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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
2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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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