AMC12 2025 B

AMC12 2025 B · Q2

AMC12 2025 B · Q2. It mainly tests Digit properties (sum of digits, divisibility tests), Sequences in number theory (remainders patterns).

Jerry wrote down the ones digit of each of the first $2025$ positive squares: $1, 4, 9, 6, 5, 6, \dots$. What is the sum of all the numbers Jerry wrote down?

杰瑞写下了前2025个正整数平方数的个位数：1, 4, 9, 6, 5, 6, \dots。杰瑞写下的所有数字之和是多少？

(A) 9025 9025

(B) 9070 9070

(D) 9115 9115

(E) 9160 9160

Answer

Correct choice: (D)

正确答案：（D）

Solution

By a modulo $10$ argument, the ones digits repeat with period $10$ in the following order: \[1,4,9,6,5,6,9,4,1,0\] The sum of the numbers can be verified to be $45$. There are $202$ periods that occur from $1$ to $2025$, and there are five extra numbers, those being $1,4,9,6,5$, corresponding to $2021,2022,2023,2024,2025$. The sum of these numbers is $25$. Hence, the total is \[202\cdot 45+25=9090+25=\boxed{\textbf{(D)}~9115}\]

通过模10论证，个位数以周期10重复，顺序为： \[1,4,9,6,5,6,9,4,1,0\] 这些数字的和为45。从1到2025有202个周期，多出5个数，对应2021至2025的1,4,9,6,5，和为25。总和为 \[202\cdot 45+25=9090+25=\boxed{\textbf{(D)}~9115}\]

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