AMC8 2022

AMC8 2022 · Q3

AMC8 2022 · Q3. It mainly tests Basic counting (rules of product/sum), Divisibility & factors.

When three positive integers $a$, $b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?

当三个正整数 $a$、$b$ 和 $c$ 相乘时，它们的乘积是 $100$。假设 $a < b < c$。可以选择这些数字有多少种方式？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (E)

正确答案：（E）

Solution

The positive divisors of $100$ are \[1,2,4,5,10,20,25,50,100.\] It is clear that $10\leq c\leq50,$ so we apply casework to $c:$ Together, the numbers $a,b,$ and $c$ can be chosen in $\boxed{\textbf{(E) } 4}$ ways.

100的正因数是 \[1,2,4,5,10,20,25,50,100.\] 显然 $10\leq c\leq50$，所以我们对 $c$ 进行分类讨论：总之，数字 $a,b,$ 和 $c$ 可以选择 $\boxed{\textbf{(E) } 4}$ 种方式。

Topics