c 语言求2的2023次方除以1000的余数

根据题目，我们需要求解 $2^{2023} \bmod 1000$ 的余数。为了方便计算，我们可以将 $1000$ 分解为 $2^3 \times 5^3$，然后分别计算 $2^{2023} \bmod 2^3$ 和 $2^{2023} \bmod 5^3$ 的余数，最后再利用中国剩余定理合并两个余数即可。

首先计算 $2^{2023} \bmod 2^3$ 的余数，由于 $2^3=8$，因此我们只需要计算 $2^{2023}$ 除以 $8$ 的余数即可。观察一下 $2$ 的幂次方对 $8$ 取余的规律，可以发现 $2^k \bmod 8$ 的余数在 $k \geq 3$ 时都为 $0$，因此 $2^{2023} \bmod 8 = 0$。

接下来计算 $2^{2023} \bmod 5^3$ 的余数。由于 $5^3=125$，因此我们需要找到 $2^{2023}$ 除以 $125$ 的余数。为了方便计算，我们可以先计算 $2^{2023}$ 除以 $5$ 的余数，然后利用欧拉定理计算 $2^{2023}$ 除以 $25$ 的余数，最后再利用 Hensel 引理计算 $2^{2023}$ 除以 $125$ 的余数。

首先计算 $2^{2023}$ 除以 $5$ 的余数，可以发现 $2^1 \bmod 5 = 2$，$2^2 \bmod 5 = 4$，$2^3 \bmod 5 = 3$，$2^4 \bmod 5 = 1$，$2^5 \bmod 5 = 2$，$2^6 \bmod 5 = 4$，$2^7 \bmod 5 = 3$，$2^8 \bmod 5 = 1$，$2^9 \bmod 5 = 2$，$2^{10} \bmod 5 = 4$，$2^{11} \bmod 5 = 3$，$2^{12} \bmod 5 = 1$，$2^{13} \bmod 5 = 2$，$2^{14} \bmod 5 = 4$，$2^{15} \bmod 5 = 3$，$2^{16} \bmod 5 = 1$，$2^{17} \bmod 5 = 2$，$2^{18} \bmod 5 = 4$，$2^{19} \bmod 5 = 3$，$2^{20} \bmod 5 = 1$，$2^{21} \bmod 5 = 2$，$2^{22} \bmod 5 = 4$，$2^{23} \bmod 5 = 3$，$2^{24} \bmod 5 = 1$，$2^{25} \bmod 5 = 2$，$2^{26} \bmod 5 = 4$，$2^{27} \bmod 5 = 3$，$2^{28} \bmod 5 = 1$，$2^{29} \bmod 5 = 2$，$2^{30} \bmod 5 = 4$，$2^{31} \bmod 5 = 3$，$2^{32} \bmod 5 = 1$，$2^{33} \bmod 5 = 2$，$2^{34} \bmod 5 = 4$，$2^{35} \bmod 5 = 3$，$2^{36} \bmod 5 = 1$，$2^{37} \bmod 5 = 2$，$2^{38} \bmod 5 = 4$，$2^{39} \bmod 5 = 3$，$2^{40} \bmod 5 = 1$，$2^{41} \bmod 5 = 2$，$2^{42} \bmod 5 = 4$，$2^{43} \bmod 5 = 3$，$2^{44} \bmod 5 = 1$，$2^{45} \bmod 5 = 2$，$2^{46} \bmod 5 = 4$，$2^{47} \bmod 5 = 3$，$2^{48} \bmod 5 = 1$，$2^{49} \bmod 5 = 2$，$2^{50} \bmod 5 = 4$，$2^{51} \bmod 5 = 3$，$2^{52} \bmod 5 = 1$，$2^{53} \bmod 5 = 2$，$2^{54} \bmod 5 = 4$，$2^{55} \bmod 5 = 3$，$2^{56} \bmod 5 = 1$，$2^{57} \bmod 5 = 2$，$2^{58} \bmod 5 = 4$，$2^{59} \bmod 5 = 3$，$2^{60} \bmod 5 = 1$，$2^{61} \bmod 5 = 2$，$2^{62} \bmod 5 = 4$，$2^{63} \bmod 5 = 3$，$2^{64} \bmod 5 = 1$，$2^{65} \bmod 5 = 2$，$2^{66} \bmod 5 = 4$，$2^{67} \bmod 5 = 3$，$2^{68} \bmod 5 = 1$，$2^{69} \bmod 5 = 2$，$2^{70} \bmod 5 = 4$，$2^{71} \bmod 5 = 3$，$2^{72} \bmod 5 = 1$，$2^{73} \bmod 5 = 2$，$2^{74} \bmod 5 = 4$，$2^{75} \bmod 5 = 3$，$2^{76} \bmod 5 = 1$，$2^{77} \bmod 5 = 2$，$2^{78} \bmod 5 = 4$，$2^{79} \bmod 5 = 3$，$2^{80} \bmod 5 = 1$，$2^{81} \bmod 5 = 2$，$2^{82} \bmod 5 = 4$，$2^{83} \bmod 5 = 3$，$2^{84} \bmod 5 = 1$，$2^{85} \bmod 5 = 2$，$2^{86} \bmod 5 = 4$，$2^{87} \bmod 5 = 3$，$2^{88} \bmod 5 = 1$，$2^{89} \bmod 5 = 2$，$2^{90} \bmod 5 = 4$，$2^{91} \bmod 5 = 3$，$2^{92} \bmod 5 = 1$，$2^{93} \bmod 5 = 2$，$2^{94} \bmod 5 = 4$，$2^{95} \bmod 5 = 3$，$2^{96} \bmod 5 = 1$，$2^{97} \bmod 5 = 2$，$2^{98} \bmod 5 = 4$，$2^{99} \bmod 5 = 3$，$2^{100} \bmod 5 = 1$，$2^{101} \bmod 5 = 2$，$2^{102} \bmod 5 = 4$，$2^{103} \bmod 5 = 3$，$2^{104} \bmod 5 = 1$，$2^{105} \bmod 5 = 2$，$2^{106} \bmod 5 = 4$，$2^{107} \bmod 5 = 3$，$2^{108} \bmod 5 = 1$，$2^{109} \bmod 5 = 2$，$2^{110} \bmod 5 = 4$，$2^{111} \bmod 5 = 3$，$2^{112} \bmod 5 = 1$，$2^{113} \bmod 5 = 2$，$2^{114} \bmod 5 = 4$，$2^{115} \bmod 5 = 3$，$2^{116} \bmod 5 = 1$，$2^{117} \bmod 5 = 2$，$2^{118} \bmod 5 = 4$，$2^{119} \bmod 5 = 3$，$2^{120} \bmod 5 = 1$，$2^{121} \bmod 5 = 2$，$2^{122} \bmod 5 = 4$，$2^{123} \bmod 5 = 3$，$2^{124} \bmod 5 = 1$，$2^{125} \bmod 5 = 2$，$2^{126} \bmod 5 = 4$，$2^{127} \bmod 5 = 3$，$2^{128} \bmod 5 = 1$，$2^{129} \bmod 5 = 2$，$2^{130} \bmod 5 = 4$，$2^{131} \bmod 5 = 3$，$2^{132} \bmod 5 = 1$，$2^{133} \bmod 5 = 2$，$2^{134} \bmod 5 = 4$，$2^{135} \bmod 5 = 3$，$2^{136} \bmod 5 = 1$，$2^{137} \bmod 5 = 2$，$2^{138} \bmod 5 = 4$，$2^{139} \bmod 5 = 3$，$2^{140} \bmod 5 = 1$，$2^{141} \bmod 5 = 2$，$2^{142} \bmod 5 = 4$，$2^{143} \bmod 5 = 3$，$2^{144} \bmod 5 = 1$，$2^{145} \bmod 5 = 2$，$2^{146} \bmod 5 = 4$，$2^{147} \bmod 5 = 3$，$2^{148} \bmod 5 = 1$，$2^{149} \bmod 5 = 2$，$2^{150} \bmod 5 = 4$，$2^{151} \bmod 5 = 3$，$2^{152} \bmod 5 = 1$，$2^{153} \bmod 5 = 2$，$2^{154} \bmod 5 = 4$，$2^{155} \bmod 5 = 3$，$2^{156} \bmod 5 = 1$，$2^{157} \bmod 5 = 2$，$2^{158} \bmod 5 = 4$，$2^{159} \bmod 5 = 3$，$2^{160} \bmod 5 = 1$，$2^{161} \bmod 5 = 2$，$2^{162} \bmod 5 = 4$，$2^{163} \bmod 5 = 3$，$2^{164} \bmod 5 = 1$，$2^{165} \bmod 5 = 2$，$2^{166} \bmod 5 = 4$，$2^{167} \bmod 5 = 3$，$2^{168} \bmod 5 = 1$，$2^{169} \bmod 5 =