AMC 8 2011 Problem 22

AMC8 2011 年第 22 题

余数与同余★★★☆☆

What is the tens digit of

7^{2011}

的十位数字是多少?

正确答案 · Correct Answer

解析 · Solution

Solution 1

Since we want the tens digit, we can find the last two digits of

7^{2011}

. We can do this by using modular arithmetic.

7^1\equiv 07 \pmod{100}.

7^2\equiv 49 \pmod{100}.

7^3\equiv 43 \pmod{100}.

7^4\equiv 01 \pmod{100}.

We can write

7^{2011}

(7^4)^{502}\times 7^3

. Using this, we can say:

7^{2011}\equiv (7^4)^{502}\times 7^3\equiv 7^3\equiv 343\equiv 43\pmod{100}.

From the above, we can conclude that the last two digits of

7^{2011}

are 43. Since they have asked us to find the tens digit, our answer is 4. -Ilovefruits

继续练习