AMC12 2014 A

AMC12 2014 A · Q19

AMC12 2014 A · Q19. It mainly tests Quadratic equations, Divisibility & factors.

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?

恰有$N$个不同的有理数$k$使得$|k|<200$且\[5x^2+kx+12=0\]至少有一个整数解$x$。$N$是多少？

(A) 6 6

(B) 12 12

(D) 48 48

(E) 78\qquad 78\qquad

Answer

Correct choice: (E)

正确答案：（E）

Solution

Factor the quadratic into \[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\] where $-n$ is our integer solution. Then, \[k = \frac{12}{n} + 5n,\] which takes rational values between $-200$ and $200$ when $|n| \leq 39$, excluding $n = 0$. This leads to an answer of $2 \cdot 39 = \boxed{\textbf{(E) } 78}$.

将二次方程因式分解为 \[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\] 其中$-n$是我们的整数解。然后， \[k = \frac{12}{n} + 5n,\] 当$|n| \leq 39$（排除$n = 0$）时，取值在$-200$和$200$之间为有理数。这导致答案为$2 \cdot 39 = \boxed{\textbf{(E) } 78}$。

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