AMC10 2017 B

AMC10 2017 B · Q14

AMC10 2017 B · Q14. It mainly tests Remainders & modular arithmetic, Digit properties (sum of digits, divisibility tests).

An integer $N$ is selected at random in the range $1 \leq N \leq 2020$. What is the probability that the remainder when $N^{16}$ is divided by 5 is 1?

随机选取整数$N$，范围$1 \leq N \leq 2020$。$N^{16}$除以5的余数为1的概率是多少？

(A) \frac{1}{5} \frac{1}{5}

(B) \frac{2}{5} \frac{2}{5}

(D) \frac{4}{5} \frac{4}{5}

(E) 1 1

Answer

Correct choice: (D)

正确答案：（D）

Solution

An integer will have a remainder of 1 when divided by 5 if and only if the units digit is either 1 or 6. The randomly selected positive integer will itself have a units digit of each of the numbers from 0 through 9 with equal probability. This digit of N alone will determine the units digit of N¹⁶. Computing the 16th power of each of these 10 digits by squaring the units digit four times yields one 0, one 5, four 1s, and four 6s. The probability is therefore 8/10 = 4/5.

整数除以5余1当且仅当个位数为1或6。随机选取的正整数的个位数为0至9各等概率。仅$N$的个位数决定$N^{16}$的个位数。通过四次平方计算每个10个数字的16次幂，得到一个0，一个5，四个1，四个6。因此概率为8/10 = 4/5。

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