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Difference between revisions of "2012 AIME I Problems/Problem 1"
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== Problem 1 ==
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== Problem==
 
Find the number of positive integers with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and <math>cba</math> are multiples of <math>4</math>.
 
Find the number of positive integers with three not necessarily distinct digits, <math>abc</math>, with <math>a \neq 0</math> and <math>c \neq 0</math> such that both <math>abc</math> and <math>cba</math> are multiples of <math>4</math>.
  
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==Solutions
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== Solution 1 ==
  
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=== Solution 1 ===
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A positive integer is divisible by <math>4</math> if and only if its last two digits are divisible by <math>4.</math> For any value of <math>b</math>, there are two possible values for <math>a</math> and <math>c</math>, since we find that if <math>b</math> is even, <math>a</math> and <math>c</math> must be either <math>4</math> or <math>8</math>, and if <math>b</math> is odd, <math>a</math> and <math>c</math> must be either <math>2</math> or <math>6</math>. There are thus <math>2 \cdot 2 = 4</math> ways to choose <math>a</math> and <math>c</math> for each <math>b,</math> and <math>10</math> ways to choose <math>b</math> since <math>b</math> can be any digit. The final answer is then <math>4 \cdot 10 = \boxed{040}</math>.
  
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A positive integer is divisible by <math>4</math> if and only if its last two digits are divisible by <math>4.</math> For any value of <math>b</math>, there are two possible values for <math>a</math> and <math>c</math>, since we find that if <math>b</math> is even, <math>a</math> and <math>c</math> must be either <math>4</math> or <math>8</math>, and if <math>b</math> is odd, <math>a</math> and <math>c</math> must be either <math>2</math> or <math>6</math>. There are thus <math>2 \cdot 2 = 4</math> ways to choose <math>a</math> and <math>c</math> for each <math>b,</math> and <math>10</math> ways to choose <math>b</math> since <math>b</math> can be any digit. The final answer is then <math>4 \cdot 10 = \boxed{040.}</math>
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== Solution 2 ==
  
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=== Solution 2 ===
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A number is divisible by four if its last two digits are divisible by 4. Thus, we require that <math> 10b + a </math> and <math> 10b + c</math> are both divisible by <math> 4 </math>. If <math> b </math> is odd, then <math> a </math> and <math> c </math> must both be <math> 2 \pmod 4 </math> meaning that <math> a </math> and <math> c </math> are <math> 2 </math> or <math> 6 </math>. If <math> b </math> is even, then <math> a </math> and <math> c </math> must be <math> 0 \pmod 4 </math> meaning that <math> a </math> and <math> c </math> are <math> 4 </math> or <math> 8 </math>. For each choice of <math> b </math> there are <math> 2 </math> choices for <math> a </math> and <math> 2 </math> for <math> c </math> for a total of <math> 10 \cdot 2 \cdot 2 = \boxed{040} </math> numbers.
  
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A number is divisible by four if its last two digits are divisible by 4. Thus, we require that <math> 10b + a </math> and <math> 10b + c</math> are both divisible by <math> 4 </math>. If <math> b </math> is odd, then <math> a </math> and <math> c </math> must both be <math> 2 \pmod 4 </math> meaning that <math> a </math> and <math> c </math> are <math> 2 </math> or <math> 6 </math>. If <math> b </math> is even, then <math> a </math> and <math> c </math> must be <math> 0 \pmod 4 </math> meaning that <math> a </math> and <math> c </math> are <math> 4 </math> or <math> 8 </math>. For each choice of <math> b </math> there are <math> 2 </math> choices for <math> a </math> and <math> 2 </math> for <math> c </math> for a total of <math> 10 \cdot 2 \cdot 2 = 040 </math> numbers.
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== Solution 3 ==
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For this number to fit the requirements <math>bc</math> and <math>ba</math> must be divisible by 4. So <math>bc = 00, 04, 08, 12, 16, ... , 92, 96</math> and so must <math>ba</math> for each two digits of <math>bc</math>. There are two possibilities for <math>ba</math> if <math>b</math> is odd and three possibilities if <math>b</math> is even. So there are <math>2^{2} \cdot 5+3^{2} \cdot 4 = 65</math> possibilities but this overcounts when <math>a</math> or <math>c = 0</math>. So when <math>bc = 00, 20, 40, 60, 80</math> and the corresponding <math>ba</math> should be removed, so <math>65 - 5 \cdot 3 = 50</math>. But we are still overcounting when <math>b</math> is even because then <math>a</math> can be 0. So the answer is <math>50 - 10 \cdot 1 = \boxed{040}</math>
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~LuisFonseca123
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== Video Solution by OmegaLearn ==
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https://youtu.be/ZhAZ1oPe5Ds?t=3235
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 +
~ pi_is_3.14
 +
 
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== Video Solutions ==
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https://artofproblemsolving.com/videos/amc/2012aimei/289
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https://www.youtube.com/watch?v=T8Ox412AkZc
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2012|n=I|before=First Problem|num-a=2}}
 
{{AIME box|year=2012|n=I|before=First Problem|num-a=2}}
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[[Category:Introductory Combinatorics Problems]]
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{{MAA Notice}}

Latest revision as of 02:59, 21 January 2023

Problem

Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \neq 0$ and $c \neq 0$ such that both $abc$ and $cba$ are multiples of $4$.

Solution 1

A positive integer is divisible by $4$ if and only if its last two digits are divisible by $4.$ For any value of $b$, there are two possible values for $a$ and $c$, since we find that if $b$ is even, $a$ and $c$ must be either $4$ or $8$, and if $b$ is odd, $a$ and $c$ must be either $2$ or $6$. There are thus $2 \cdot 2 = 4$ ways to choose $a$ and $c$ for each $b,$ and $10$ ways to choose $b$ since $b$ can be any digit. The final answer is then $4 \cdot 10 = \boxed{040}$.

Solution 2

A number is divisible by four if its last two digits are divisible by 4. Thus, we require that $10b + a$ and $10b + c$ are both divisible by $4$. If $b$ is odd, then $a$ and $c$ must both be $2 \pmod 4$ meaning that $a$ and $c$ are $2$ or $6$. If $b$ is even, then $a$ and $c$ must be $0 \pmod 4$ meaning that $a$ and $c$ are $4$ or $8$. For each choice of $b$ there are $2$ choices for $a$ and $2$ for $c$ for a total of $10 \cdot 2 \cdot 2 = \boxed{040}$ numbers.

Solution 3

For this number to fit the requirements $bc$ and $ba$ must be divisible by 4. So $bc = 00, 04, 08, 12, 16, ... , 92, 96$ and so must $ba$ for each two digits of $bc$. There are two possibilities for $ba$ if $b$ is odd and three possibilities if $b$ is even. So there are $2^{2} \cdot 5+3^{2} \cdot 4 = 65$ possibilities but this overcounts when $a$ or $c = 0$. So when $bc = 00, 20, 40, 60, 80$ and the corresponding $ba$ should be removed, so $65 - 5 \cdot 3 = 50$. But we are still overcounting when $b$ is even because then $a$ can be 0. So the answer is $50 - 10 \cdot 1 = \boxed{040}$

~LuisFonseca123

Video Solution by OmegaLearn

https://youtu.be/ZhAZ1oPe5Ds?t=3235

~ pi_is_3.14

Video Solutions

https://artofproblemsolving.com/videos/amc/2012aimei/289

https://www.youtube.com/watch?v=T8Ox412AkZc

See also

2012 AIME I (Problems • Answer Key • Resources)
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