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Difference between revisions of "2023 AIME I Problems/Problem 7"
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Unofficial Solution: We realize that any such number (mod 2) and (mod 4) must have the same parity, and its values (mod 3) and (mod 6) must have a absolute value difference of 3. Thus the only possibilities for the sequence of mods are 1,2,3,4,5 1,2,3,0,5 and 0,1,2,3,4. Using CRT and summing we get 049.
 
Unofficial Solution: We realize that any such number (mod 2) and (mod 4) must have the same parity, and its values (mod 3) and (mod 6) must have a absolute value difference of 3. Thus the only possibilities for the sequence of mods are 1,2,3,4,5 1,2,3,0,5 and 0,1,2,3,4. Using CRT and summing we get 049.
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==Solution==
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\textbf{Case 0:} <math>{\rm Rem} \ \left( n, 6 \right) = 0</math>.
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We have <math>{\rm Rem} \ \left( n, 2 \right) = 0</math>. This violates the condition that <math>n</math> is extra-distinct.
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Therefore, this case has no solution.
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\textbf{Case 1:} <math>{\rm Rem} \ \left( n, 6 \right) = 1</math>.
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We have <math>{\rm Rem} \ \left( n, 2 \right) = 1</math>. This violates the condition that <math>n</math> is extra-distinct.
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Therefore, this case has no solution.
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\textbf{Case 2:} <math>{\rm Rem} \ \left( n, 6 \right) = 2</math>.
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We have <math>{\rm Rem} \ \left( n, 3 \right) = 2</math>. This violates the condition that <math>n</math> is extra-distinct.
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Therefore, this case has no solution.
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\textbf{Case 3:} <math>{\rm Rem} \ \left( n, 6 \right) = 3</math>.
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The condition <math>{\rm Rem} \ \left( n, 6 \right) = 3</math> implies <math>{\rm Rem} \ \left( n, 2 \right) = 1</math>, <math>{\rm Rem} \ \left( n, 3 \right) = 0</math>.
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Because <math>n</math> is extra-distinct, <math>{\rm Rem} \ \left( n, l \right)</math> for <math>l \in \left\{ 2, 3, 4 \right\}</math> is a permutation of <math>\left\{ 0, 1 ,2 \right\}</math>.
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Thus, <math>{\rm Rem} \ \left( n, 4 \right) = 2</math>.
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However, <math>{\rm Rem} \ \left( n, 4 \right) = 2</math> conflicts <math>{\rm Rem} \ \left( n, 2 \right) = 1</math>.
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Therefore, this case has no solution.
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\textbf{Case 4:} <math>{\rm Rem} \ \left( n, 6 \right) = 4</math>.
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The condition <math>{\rm Rem} \ \left( n, 6 \right) = 4</math> implies <math>{\rm Rem} \ \left( n, 2 \right) = 0</math> and <math>{\rm Rem} \ \left( n, 3 \right) = 1</math>.
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Because <math>n</math> is extra-distinct, <math>{\rm Rem} \ \left( n, l \right)</math> for <math>l \in \left\{ 2, 3, 4 , 5 \right\}</math> is a permutation of <math>\left\{ 0, 1 ,2 , 3 \right\}</math>.
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Because <math>{\rm Rem} \ \left( n, 2 \right) = 0</math>, we must have <math>{\rm Rem} \ \left( n, 4 \right) = 2</math>. Hence, <math>{\rm Rem} \ \left( n, 5 \right) = 3</math>.
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Hence, <math>n \equiv -2 \pmod{{\rm lcm} \left( 4, 5 , 6 \right)}</math>.
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Hence, <math>n \equiv - 2 \pmod{60}</math>.
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We have <math>1000 = 60 \cdot 16 + 40</math>.
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Therefore, the number extra-distinct <math>n</math> in this case is 16.
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\textbf{Case 5:} <math>{\rm Rem} \ \left( n, 6 \right) = 5</math>.
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The condition <math>{\rm Rem} \ \left( n, 6 \right) = 5</math> implies <math>{\rm Rem} \ \left( n, 2 \right) = 1</math> and <math>{\rm Rem} \ \left( n, 3 \right) = 2</math>.
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Because <math>n</math> is extra-distinct, <math>{\rm Rem} \ \left( n, 4 \right)</math> and <math>{\rm Rem} \ \left( n, 5 \right)</math> are two distinct numbers in <math>\left\{ 0, 3, 4 \right\}</math>.
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Because <math>{\rm Rem} \ \left( n, 4 \right) \leq 3</math> and <math>n</math> is odd, we have <math>{\rm Rem} \ \left( n, 4 \right) = 3</math>.
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Hence, <math>{\rm Rem} \ \left( n, 5 \right) = 0</math> or 4.
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\textbf{Case 5.1:} <math>{\rm Rem} \ \left( n, 6 \right) = 5</math>, <math>{\rm Rem} \ \left( n, 4 \right) = 3</math>, <math>{\rm Rem} \ \left( n, 5 \right) = 0</math>.
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We have <math>n \equiv 35 \pmod{60}</math>.
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We have <math>1000 = 60 \cdot 16 + 40</math>.
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Therefore, the number extra-distinct <math>n</math> in this subcase is 17.
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\textbf{Case 5.2:} <math>{\rm Rem} \ \left( n, 6 \right) = 5</math>, <math>{\rm Rem} \ \left( n, 4 \right) = 3</math>, <math>{\rm Rem} \ \left( n, 5 \right) = 4</math>.
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<math>n \equiv - 1 \pmod{60}</math>.
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We have <math>1000 = 60 \cdot 16 + 40</math>.
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Therefore, the number extra-distinct <math>n</math> in this subcase is 16.
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Putting all cases together, the total number of extra-distinct <math>n</math> is <math>16 + 17 + 16 = \boxed{\textbf{(049) }}</math>.
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 13:39, 8 February 2023

Unofficial problem statement: Find the number of positive integers from 1 to 1000 that have different mods in mod 2,3,4,5, and 6.

Unofficial Solution: We realize that any such number (mod 2) and (mod 4) must have the same parity, and its values (mod 3) and (mod 6) must have a absolute value difference of 3. Thus the only possibilities for the sequence of mods are 1,2,3,4,5 1,2,3,0,5 and 0,1,2,3,4. Using CRT and summing we get 049.

Solution

\textbf{Case 0:} ${\rm Rem} \ \left( n, 6 \right) = 0$.

We have ${\rm Rem} \ \left( n, 2 \right) = 0$. This violates the condition that $n$ is extra-distinct. Therefore, this case has no solution.

\textbf{Case 1:} ${\rm Rem} \ \left( n, 6 \right) = 1$.

We have ${\rm Rem} \ \left( n, 2 \right) = 1$. This violates the condition that $n$ is extra-distinct. Therefore, this case has no solution.

\textbf{Case 2:} ${\rm Rem} \ \left( n, 6 \right) = 2$.

We have ${\rm Rem} \ \left( n, 3 \right) = 2$. This violates the condition that $n$ is extra-distinct. Therefore, this case has no solution.

\textbf{Case 3:} ${\rm Rem} \ \left( n, 6 \right) = 3$.

The condition ${\rm Rem} \ \left( n, 6 \right) = 3$ implies ${\rm Rem} \ \left( n, 2 \right) = 1$, ${\rm Rem} \ \left( n, 3 \right) = 0$.

Because $n$ is extra-distinct, ${\rm Rem} \ \left( n, l \right)$ for $l \in \left\{ 2, 3, 4 \right\}$ is a permutation of $\left\{ 0, 1 ,2 \right\}$. Thus, ${\rm Rem} \ \left( n, 4 \right) = 2$.

However, ${\rm Rem} \ \left( n, 4 \right) = 2$ conflicts ${\rm Rem} \ \left( n, 2 \right) = 1$. Therefore, this case has no solution.

\textbf{Case 4:} ${\rm Rem} \ \left( n, 6 \right) = 4$.

The condition ${\rm Rem} \ \left( n, 6 \right) = 4$ implies ${\rm Rem} \ \left( n, 2 \right) = 0$ and ${\rm Rem} \ \left( n, 3 \right) = 1$.

Because $n$ is extra-distinct, ${\rm Rem} \ \left( n, l \right)$ for $l \in \left\{ 2, 3, 4 , 5 \right\}$ is a permutation of $\left\{ 0, 1 ,2 , 3 \right\}$.

Because ${\rm Rem} \ \left( n, 2 \right) = 0$, we must have ${\rm Rem} \ \left( n, 4 \right) = 2$. Hence, ${\rm Rem} \ \left( n, 5 \right) = 3$.

Hence, $n \equiv -2 \pmod{{\rm lcm} \left( 4, 5 , 6 \right)}$. Hence, $n \equiv - 2 \pmod{60}$.

We have $1000 = 60 \cdot 16 + 40$. Therefore, the number extra-distinct $n$ in this case is 16.

\textbf{Case 5:} ${\rm Rem} \ \left( n, 6 \right) = 5$.

The condition ${\rm Rem} \ \left( n, 6 \right) = 5$ implies ${\rm Rem} \ \left( n, 2 \right) = 1$ and ${\rm Rem} \ \left( n, 3 \right) = 2$.

Because $n$ is extra-distinct, ${\rm Rem} \ \left( n, 4 \right)$ and ${\rm Rem} \ \left( n, 5 \right)$ are two distinct numbers in $\left\{ 0, 3, 4 \right\}$. Because ${\rm Rem} \ \left( n, 4 \right) \leq 3$ and $n$ is odd, we have ${\rm Rem} \ \left( n, 4 \right) = 3$. Hence, ${\rm Rem} \ \left( n, 5 \right) = 0$ or 4.

\textbf{Case 5.1:} ${\rm Rem} \ \left( n, 6 \right) = 5$, ${\rm Rem} \ \left( n, 4 \right) = 3$, ${\rm Rem} \ \left( n, 5 \right) = 0$.

We have $n \equiv 35 \pmod{60}$.

We have $1000 = 60 \cdot 16 + 40$. Therefore, the number extra-distinct $n$ in this subcase is 17.

\textbf{Case 5.2:} ${\rm Rem} \ \left( n, 6 \right) = 5$, ${\rm Rem} \ \left( n, 4 \right) = 3$, ${\rm Rem} \ \left( n, 5 \right) = 4$.

$n \equiv - 1 \pmod{60}$.

We have $1000 = 60 \cdot 16 + 40$. Therefore, the number extra-distinct $n$ in this subcase is 16.

Putting all cases together, the total number of extra-distinct $n$ is $16 + 17 + 16 = \boxed{\textbf{(049) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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