AMC 10 Daily Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 12 Medium

In the coordinate system, the x and y coordinates of the intersection point between lines y=x-k and y=kx+2 are both integers. How many possible values of k are there?

  • A.

    4

  • B.

    5

  • C.

    6

  • D.

    7

  • E.

    8

Answer:A

Construct the system \begin{cases}y=x-k ①\\ y=kx+2② \\ \end{cases}.

Plug the first equation into the second equation, we have x\left( 1-k \right)=k+2.

It is clear that 1-k\ne 0, so we have x=\frac{k+2}{1-k}=-1+\frac{3}{1-k}.

When 1-k=\pm 3 or \pm 1, x is an integer, and now y is also an integer.

Therefore, k=\{4,2,0,-2\}, four of them.

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