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Difference between revisions of "2022 AMC 10B Problems/Problem 8"
 
(14 intermediate revisions by 7 users not shown)
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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 1==
+
==Solution 1 (Casework)==
 
We apply casework to this problem. The only sets that contain two multiples of seven are those for which:
 
We apply casework to this problem. The only sets that contain two multiples of seven are those for which:
  
Line 30: Line 29:
 
~MRENTHUSIASM
 
~MRENTHUSIASM
  
−
==Solution 2==
+
==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>2</math> multiples of <math>7</math>. Calculating this pattern further, we can see (reasonably) that it repeats for each <math>7</math> sets.
 +
We see that the pattern for the number of multiples per <math>7</math> sets goes: <math>1,1,2,1,2,1,2.</math> So, for every <math>7</math> sets, there are three sets with <math>2</math> multiples of <math>7</math>. We calculate <math>\left\lfloor\frac{100}{7}\right\rfloor</math> and multiply that by <math>3</math>. (We also disregard the remainder of <math>2</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) ProProtractor
 +
 
 +
==Solution 3 (Fastest)==
 
Each set contains exactly <math>1</math> or <math>2</math> multiples of <math>7</math>.
 
Each set contains exactly <math>1</math> or <math>2</math> multiples of <math>7</math>.
  
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~BrandonZhang202415
 
~BrandonZhang202415
  
−
==Solution 3==
+
==Video Solution (🚀Under 3 min🚀)==
−
We find a pattern.  
+
https://youtu.be/PdyKJ1p9Y2w
−
<cmath>\begin{align*}
+
 
−
&\{1,2,3,\ldots,10\}, \\
+
~Education, the Study of Everything
−
&\{11,12,13,\ldots,20\},\\
+
 
−
&\{21,22,23,\ldots,30\},\\
+
==Video Solution(1-16)==
−
&\vdots\\
+
https://youtu.be/SCwQ9jUfr0g
−
&\{991,992,993,\ldots,1000\}.
+
 
−
\end{align*}</cmath>
+
~~Hayabusa1
−
Through quick listing <math>(7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98)</math>, we can figure out that the first set has 1 multiple of 7. The second set 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 2 multiples of 7. The eighth set has 1 multiple of 7. The ninth set has 1 multiples of 7. The tenth set has 2 multiples of 7.
 
−
We see that the pattern for the number of multiples per set goes: <math>1,1,2,1,2,1,2,1,1,2.</math> We can reasonably conclude that the pattern <math>1,1,2,1,2,1,2</math> repeats every 7 times. So, for every 7 sets, there are three multiples of 7. We calculate the floor of 100/7 and multiply that by 3 (we disregard the remainder of two since it doesn't add any extra sets with 2 multiples of 7). We get <math>14*3= \boxed{\textbf{(B) }42}</math>
 
  
 +
==Video Solution by Interstigation==
 +
https://youtu.be/_KNR0JV5rdI?t=884
  
 
== See Also ==
 
== See Also ==

Latest revision as of 17:40, 11 November 2023

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 $2$ multiples of $7$. Calculating this pattern further, we can see (reasonably) that it repeats for each $7$ sets. We see that the pattern for the number of multiples per $7$ sets goes: $1,1,2,1,2,1,2.$ So, for every $7$ sets, there are three sets with $2$ multiples of $7$. We calculate $\left\lfloor\frac{100}{7}\right\rfloor$ and multiply that by $3$. (We also disregard the remainder of $2$ since it doesn't add any extra sets with $2$ multiples of $7$.). We get $14\cdot3= \boxed{\textbf{(B) }42}$.

~(edited by) ProProtractor

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

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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