AMC10 2002 A

AMC10 2002 A · Q4

AMC10 2002 A · Q4. It mainly tests Linear inequalities.

For how many positive integers $m$ does there exist at least one positive integer $n$ such that $m \cdot n \leq m + n$?

有且仅有有多少个正整数 $m$，使得存在至少一个正整数 $n$ 满足 $m \cdot n \leq m + n$？

(A) 4 4

(B) 6 6

(D) 12 12

(E) infinitely many 无数个

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) When $n=1$, the inequality becomes $m \le 1+m$, which is satisfied by all integers $m$. Thus, there are infinitely many of the desired values of $m$.

（E）当 $n=1$ 时，不等式变为 $m \le 1+m$，对所有整数 $m$ 都成立。因此，满足条件的 $m$ 有无穷多个。

Topics

AL.003 Linear inequalities