3's and 7's

Number Theory Level 5

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.

1 solution

Alex Delhumeau
Oct 12, 2015

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.

3's and 7's

The answer is 2072.

1 solution

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