AMC8 2025

AMC8 2025 · Q16

AMC8 2025 · Q16. It mainly tests Basic counting (rules of product/sum), Digit properties (sum of digits, divisibility tests).

Five distinct integers from $1$ to $10$ are chosen, and five distinct integers from $11$ to $20$ are chosen. No two numbers differ by exactly $10$. What is the sum of the ten chosen numbers?

从$1$到$10$中选择5个不同的整数，从$11$到$20$中选择5个不同的整数。没有任何两个数相差恰好$10$。所选十个数的和是多少？

(A) \ 95 \ 95

(B) \ 100 \ 100

(D) \ 110 \ 110

(E) \ 115 \ 115

Answer

Correct choice: (C)

正确答案：（C）

Solution

Note that for no two numbers to differ by $10$, every number chosen must have a different units digit. To make computations easier, we can choose $(1, 2, 3, 4, 5)$ from the first group and $(16, 17, 18, 19, 20)$ from the second group. Then the sum evaluates to $1+2+3+4+5+16+17+18+19+20 = \boxed{\text{(C) 105}}$.

注意到没有两个数相差$10$，意味着每个选择的数必须有不同的个位数。为了计算更方便，我们可以从第一组选择$(1, 2, 3, 4, 5)$，从第二组选择$(16, 17, 18, 19, 20)$。则和为$1+2+3+4+5+16+17+18+19+20 = \boxed{\text{(C) 105}}$。

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