3 and 7 are coprime. For some integer
$(x, y)$
, the four integer satisfied
$3x+7y=1$
For
$i = 1, 2, 3, \dots$
, let
$x_i$
be the sequence of
**
positive
**
integer
$x$
such that
$x_i < x_{i+1}$
, each
$x_i$
produce the sequence
$y_i$
of integer
$y$
satisfying the equation above, and
$x_1$
is the
**
least positive
**
integer
$x$

Find the value of $\displaystyle \sum_{i=1}^{37}|y_i|$ .

The answer is 2072.

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The given expression can be rewritten as $y=\frac{-3x+1}{7}$ , showing that for $y$ to be an integer, $-3x+1 \equiv 0\pmod{7} \Rightarrow x\equiv5\pmod{7} \Rightarrow x = 5+7n.$ Furthermore, we can say $n \geq 0$ since $x \geq 0$ .

Substitution now gives $y=\frac{-3(5+7n)+1}{7} \Rightarrow y = -2 - 3n$ . Taking the sum from $0$ to $36$ , we now get $-2072$ , which we make positive to get the answer.