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Suppose positive integers $a$ and $b$ are relatively prime and when $b$ is divided by $a$, $4$ and $7$ are their remainder and quotient, respectively, Let $a_1,a_2,a_3,a_4,\ldots$ be all the numbers $a$ (in ascending order) that satisfy the above conditions. Find $a_{2014}$.

We have $b = 7a+4$ and so by the Euclidean Algorithm, $\gcd(a,b) = \gcd(a,4)$. Thus, $a$ is odd and is one of $5,7,\ldots$. The $2014$-th such number is $5+2013 \cdot 2 = 4031$.

The answer key says the answer is $28221$, which is $7 \cdot 4031+4$. How are they getting that?

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It seems this is simply an error: they are giving the $2014$th value of $b$ instead of $a$. I would guess that the error is actually in the problem statement rather than the answer, and that it meant to say "when $a$ is divided by $b$" instead of "when $b$ is divided by $a$".

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This is clearly the corresponding value of $b$. Could it be a typo in the question? Maybe, $a$ and $b$ were switched in the definition? I mean, $a$ is divided by $b$ rather than the other way around.

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It looks like error to me as well. Perhaps they are looking for sequence:

$b_1, b_2, b_3, ...$ then we have arithmetic progress starting from second term with the difference of $14$:

$11, \dfrac{25}{3}, \dfrac{39}{5}, \dfrac{53}{7}, ...$

$28221=25+14 \cdot 2014$

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