AMC12 2005 A

AMC12 2005 A · Q12

AMC12 2005 A · Q12. It mainly tests Coordinate geometry, Primes & prime factorization.

A line passes through $A(1,1)$ and $B(100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$?

一条直线经过点 $A(1,1)$ 和 $B(100,1000)$。在这条直线上，严格位于 $A$ 和 $B$ 之间且具有整数坐标的其它点有多少个？

(A) 0 0

(B) 2 2

(D) 8 8

(E) 9 9

Answer

Correct choice: (D)

正确答案：（D）

Solution

For convenience’s sake, we can transform $A$ to the origin and $B$ to $(99,999)$ (this does not change the problem). The line $AB$ has the equation $y = \frac{999}{99}x = \frac{111}{11}x$. The coordinates are integers if $11|x$, so the values of $x$ are $11, 22 \ldots 88$, with a total of $8\implies \boxed{\mathrm{(D)}}$ coordinates.

为方便起见，将 $A$ 平移到原点、将 $B$ 平移到 $(99,999)$（这不改变问题）。直线 $AB$ 的方程为 $y = \frac{999}{99}x = \frac{111}{11}x$。当且仅当 $11|x$ 时坐标为整数，因此 $x$ 可取 $11, 22 \ldots 88$，共 $8\implies \boxed{\mathrm{(D)}}$ 个点。

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