AMC10 2003 A

AMC10 2003 A · Q16

AMC10 2003 A · Q16. It mainly tests Digit properties (sum of digits, divisibility tests), Powers & residues.

What is the units digit of $13^{2003}$?

$13^{2003}$ 的单位数字是多少？

(A) 1 1

(B) 3 3

(D) 8 8

(E) 9 9

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Powers of 13 have the same units digit as the corresponding powers of 3; and $3^1=3,\quad 3^2=9,\quad 3^3=27,\quad 3^4=81,$ and $3^5=243.$ Since the units digit of $3^1$ is the same as the units digit of $3^5$, units digits of powers of 3 cycle through $3,9,7,$ and $1$. Hence the units digit of $3^{2000}$ is $1$, so the units digit of $3^{2003}$ is $7$. The same is true of the units digit of $13^{2003}$.

（C）$13$ 的幂与对应的 $3$ 的幂具有相同的个位数字；并且 $3^1=3,\quad 3^2=9,\quad 3^3=27,\quad 3^4=81,$ 且 $3^5=243.$ 由于 $3^1$ 的个位数与 $3^5$ 的个位数相同，$3$ 的各次幂的个位数按 $3,9,7,1$ 循环。因此 $3^{2000}$ 的个位数是 $1$，所以 $3^{2003}$ 的个位数是 $7$。$13^{2003}$ 的个位数也同样如此。

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