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Difference between revisions of "2022 AIME I Problems/Problem 2"

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Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits.
 
Find the three-digit positive integer <math>\underline{a}\,\underline{b}\,\underline{c}</math> whose representation in base nine is <math>\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},</math> where <math>a,</math> <math>b,</math> and <math>c</math> are (not necessarily distinct) digits.
  
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== Solution ==
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== Solution 1 ==
  
 
We are given that <cmath>100a + 10b + c = 81b + 9c + a,</cmath> which rearranges to <cmath>99a = 71b + 8c.</cmath>
 
We are given that <cmath>100a + 10b + c = 81b + 9c + a,</cmath> which rearranges to <cmath>99a = 71b + 8c.</cmath>
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~MRENTHUSIASM
 
~MRENTHUSIASM
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== Solution 2 ==
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As shown in Solution 1, we get <math>99a = 71b+8c</math>.
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Note that <math>99</math> and <math>71</math> are large numbers comparatively to <math>8</math>, so we hypothesize that <math>a</math> and <math>b</math> are equal and <math>8c</math> fills the gap between them. The difference between <math>99</math> and <math>71</math> is <math>28</math>, which is a multiple of <math>4</math>. So, if we multiply this by <math>2</math>, it will be a multiple of <math>8</math> and thus the gap can be filled. Therefore, the only solution is <math>(a,b,c)=(2,2,7)</math>, and the answer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}</math>.
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~KingRavi
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== Solution 3 ==
 +
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As shown in Solution 1, we get <math>99a = 71b+8c.</math>
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We list a few multiples of <math>99</math> out: <cmath>99,198,297,396.</cmath>
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Of course, <math>99</math> can't be made of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>28</math>, which can't be made of <math>8</math>'s either. So <math>99</math> doesn't work.
 +
<math>198</math> can't be made up of just <math>8</math>'s. If we use one <math>71</math>, we get a remainder of <math>127</math>, which can't be made of <math>8</math>'s. If we use two <math>71</math>'s, we get a remainder of <math>56</math>, which can be made of <math>8</math>'s.
 +
Therefore we get <math>99\cdot2=71\cdot2+8\cdot7</math> so <math>a=2,b=2,</math> and <math>c=7</math>. Plugging this back into the original problem shows that this answer is indeed correct. Therefore, <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.</math>
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~Technodoggo
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== Solution 4 ==
 +
As shown in Solution 1, we get <math>99a = 71b+8c</math>.
 +
 +
We can see that <math>99</math> is <math>28</math> larger than <math>71</math>, and we have an <math>8c</math>. We can clearly see that <math>56</math> is a multiple of <math>8</math>, and any larger than <math>56</math> would result in <math>c</math> being larger than <math>9</math>. Therefore, our only solution is <math>a = 2, b = 2, c = 7</math>. Our answer is <math>\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}</math>.
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~Arcticturn
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== Solution 5 ==
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As shown in Solution 1, we get <math>99a = 71b+8c,</math> which rearranges to <cmath>99(a – b) = 8c – 28 b = 4(2c – 7b) \le 4(2\cdot 9 - 0 ) = 72.</cmath>
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So <math>a=b, 2c = 7b \implies c=7, b=2,a=2.</math>
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'''vladimir.shelomovskii@gmail.com, vvsss'''
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== Solution 6 ==
 +
 +
As shown in Solution 1, we have that <math>99a = 71b + 8c</math>.
 +
 +
Note that by the divisibility rule for <math>9</math>, we have <math>a+b+c \equiv a \pmod{9}</math>.  Since <math>b</math> and <math>c</math> are base-<math>9</math> digits, we can say that <math>b+c = 0</math> or <math>b+c=9</math>.  The former possibility can be easily eliminated, and thus <math>b+c=9</math>.  Next, we write the equation from Solution 1 as <math>99a = 63b + 8(b+c)</math>, and dividing this by <math>9</math> gives <math>11a = 7b+8</math>.  Taking both sides modulo <math>7</math>, we have <math>4a \equiv 1 \pmod{7}</math>.  Multiplying both sides by <math>2</math> gives <math>a\equiv 2 \pmod{7}</math>, which implies <math>a=2</math>.  From here, we can find that <math>b=2</math> and <math>c=7</math>, giving an answer of <math>\boxed{227}</math>.
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~Sedro
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== Video Solution by OmegaLearn ==
 +
https://youtu.be/SCGzEOOICr4?t=340
 +
 +
~ pi_is_3.14
  
 
==Video Solution (Mathematical Dexterity)==
 
==Video Solution (Mathematical Dexterity)==
 
https://www.youtube.com/watch?v=z5Y4bT5rL-s
 
https://www.youtube.com/watch?v=z5Y4bT5rL-s
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=CwSkAHR3AcM
 +
 +
~Steven Chen (www.professorchenedu.com)
 +
 +
== Video Solution ==
 +
 +
https://youtu.be/MJ_M-xvwHLk?t=392
 +
 +
~ThePuzzlr
 +
 +
==Video Solution by MRENTHUSIASM (English & Chinese)==
 +
https://www.youtube.com/watch?v=v4tHtlcD9ww&t=360s&ab_channel=MRENTHUSIASM
 +
 +
~MRENTHUSIASM
 +
 +
== Video Solution ==
 +
https://youtu.be/YcZzxez-j-c
 +
 +
~AMC & AIME Training
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=I|num-b=1|num-a=3}}
 
{{AIME box|year=2022|n=I|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:05, 2 February 2024

Problem

Find the three-digit positive integer $\underline{a}\,\underline{b}\,\underline{c}$ whose representation in base nine is $\underline{b}\,\underline{c}\,\underline{a}_{\,\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits.

Solution 1

We are given that \[100a + 10b + c = 81b + 9c + a,\] which rearranges to \[99a = 71b + 8c.\] Taking both sides modulo $71,$ we have \begin{align*} 28a &\equiv 8c \pmod{71} \\ 7a &\equiv 2c \pmod{71}. \end{align*} The only solution occurs at $(a,c)=(2,7),$ from which $b=2.$

Therefore, the requested three-digit positive integer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.$

~MRENTHUSIASM

Solution 2

As shown in Solution 1, we get $99a = 71b+8c$.

Note that $99$ and $71$ are large numbers comparatively to $8$, so we hypothesize that $a$ and $b$ are equal and $8c$ fills the gap between them. The difference between $99$ and $71$ is $28$, which is a multiple of $4$. So, if we multiply this by $2$, it will be a multiple of $8$ and thus the gap can be filled. Therefore, the only solution is $(a,b,c)=(2,2,7)$, and the answer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}$.

~KingRavi

Solution 3

As shown in Solution 1, we get $99a = 71b+8c.$

We list a few multiples of $99$ out: \[99,198,297,396.\] Of course, $99$ can't be made of just $8$'s. If we use one $71$, we get a remainder of $28$, which can't be made of $8$'s either. So $99$ doesn't work. $198$ can't be made up of just $8$'s. If we use one $71$, we get a remainder of $127$, which can't be made of $8$'s. If we use two $71$'s, we get a remainder of $56$, which can be made of $8$'s. Therefore we get $99\cdot2=71\cdot2+8\cdot7$ so $a=2,b=2,$ and $c=7$. Plugging this back into the original problem shows that this answer is indeed correct. Therefore, $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}.$

~Technodoggo

Solution 4

As shown in Solution 1, we get $99a = 71b+8c$.

We can see that $99$ is $28$ larger than $71$, and we have an $8c$. We can clearly see that $56$ is a multiple of $8$, and any larger than $56$ would result in $c$ being larger than $9$. Therefore, our only solution is $a = 2, b = 2, c = 7$. Our answer is $\underline{a}\,\underline{b}\,\underline{c}=\boxed{227}$.

~Arcticturn

Solution 5

As shown in Solution 1, we get $99a = 71b+8c,$ which rearranges to \[99(a – b) = 8c – 28 b = 4(2c – 7b) \le 4(2\cdot 9 - 0 ) = 72.\] So $a=b, 2c = 7b \implies c=7, b=2,a=2.$

vladimir.shelomovskii@gmail.com, vvsss

Solution 6

As shown in Solution 1, we have that $99a = 71b + 8c$.

Note that by the divisibility rule for $9$, we have $a+b+c \equiv a \pmod{9}$. Since $b$ and $c$ are base-$9$ digits, we can say that $b+c = 0$ or $b+c=9$. The former possibility can be easily eliminated, and thus $b+c=9$. Next, we write the equation from Solution 1 as $99a = 63b + 8(b+c)$, and dividing this by $9$ gives $11a = 7b+8$. Taking both sides modulo $7$, we have $4a \equiv 1 \pmod{7}$. Multiplying both sides by $2$ gives $a\equiv 2 \pmod{7}$, which implies $a=2$. From here, we can find that $b=2$ and $c=7$, giving an answer of $\boxed{227}$.

~Sedro

Video Solution by OmegaLearn

https://youtu.be/SCGzEOOICr4?t=340

~ pi_is_3.14

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=z5Y4bT5rL-s

Video Solution

https://www.youtube.com/watch?v=CwSkAHR3AcM

~Steven Chen (www.professorchenedu.com)

Video Solution

https://youtu.be/MJ_M-xvwHLk?t=392

~ThePuzzlr

Video Solution by MRENTHUSIASM (English & Chinese)

https://www.youtube.com/watch?v=v4tHtlcD9ww&t=360s&ab_channel=MRENTHUSIASM

~MRENTHUSIASM

Video Solution

https://youtu.be/YcZzxez-j-c

~AMC & AIME Training

See Also

2022 AIME I (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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