挑战程序竞赛系列（42）：4.1模运算的世界（4）

详细代码可以fork下Github上leetcode项目，不定期更新。

练习题如下：

• POJ 2720: Last Digits

POJ 2720: Last Digits

此题的难点在于如何解决溢出问题，这里用到数论的一个知识点，没办法只能照葫芦画瓢。

具体利用了欧拉函数，算法如下：

    public long f(int b, int i, int mod){        if (i == 0) return 1;        else if (mod == 1) return 0;        else if (fbx[b][i] < 0){            int euler = (int) euler_phi(mod);            return pow(b, f(b, i - 1, euler) + euler, mod);        }        else{            return fbx[b][i] % mod;        }    }

好了，f函数编写完成之后，我们用快速幂来实现pow函数，先打表fbx把能算的算出来。

最后再用一个cache把已经算出的值缓存起来，因为后续输出变动的原因在于n的变化。

代码如下：

import java.io.BufferedReader;import java.io.File;import java.io.FileInputStream;import java.io.IOException;import java.io.InputStream;import java.io.InputStreamReader;import java.io.PrintWriter;import java.util.Arrays;import java.util.StringTokenizer;public class Main{    String INPUT = "./data/judge/201708/P2720.txt";    public static void main(String[] args) throws IOException {        new Main().run();    }    static final int MAX_N = 101;    int[][] fbx = new int[MAX_N][MAX_N];    int[] ten_power = new int[8];    public int power_of(int x, int y){        if (y < 0) return -1;        int res = 1;        while (y > 0){            if ((y & 1) != 0){                res = res * x;                if (res >= ten_power[7] || res <= 0) return -1;            }            y >>= 1;            x = x * x;        }        return res;    }    public void prepare(){        ten_power[0] = 1;        for (int i = 1; i < 8; ++i){            ten_power[i] = ten_power[i - 1] * 10;        }        for (int i = 0; i < MAX_N; ++i) Arrays.fill(fbx[i], -1);        for (int i = 1; i < MAX_N; ++i){            fbx[i][0] = 1;            for (int j = 1; j < MAX_N; ++j){                fbx[i][j] = power_of(i, fbx[i][j - 1]);            }        }    }    int[][] cache = new int[MAX_N][MAX_N];    void solve() {        prepare();        for (int i = 0; i < MAX_N; ++i) Arrays.fill(cache[i], -1);        while (true){            int b = ni();            if (b == 0) break;            int i = ni();            int n = ni();            int ans = 0;            if (cache[b][i] < 0){                if (b == 1){                    ans = 1;                }                else ans = (int) f(b, i, ten_power[7]);                cache[b][i] = ans;            }            ans = cache[b][i] % ten_power[n];            StringBuilder sb = new StringBuilder();            for (int j = 0; j < n - String.valueOf(ans).length(); ++j){                sb.append("0");            }            out.println(sb.toString() + ans);        }    }    public long euler_phi(int m){        long res = m;        for (int i = 2; i <= m / i; ++i){            if (m % i == 0){                res = res / i * (i - 1);                for (; m % i == 0; m /= i);            }        }        if (m != 1){            res = res / m * (m - 1);        }        return res;    }    public long f(int b, int i, int mod){        if (i == 0) return 1;        else if (mod == 1) return 0;        else if (fbx[b][i] < 0){            int euler = (int) euler_phi(mod);            return pow(b, f(b, i - 1, euler) + euler, mod);        }        else{            return fbx[b][i] % mod;        }    }    public long pow(long base, long n, long mod){        long res = 1;        while (n > 0){            if ((n & 1) != 0){                res = res * base % mod;            }            n >>= 1;            base = base * base % mod;        }        return res;    }    FastScanner in;    PrintWriter out;    void run() throws IOException {        boolean oj;        try {            oj = ! System.getProperty("user.dir").equals("F:\\java_workspace\\leetcode");        } catch (Exception e) {            oj = System.getProperty("ONLINE_JUDGE") != null;        }        InputStream is = oj ? System.in : new FileInputStream(new File(INPUT));        in = new FastScanner(is);        out = new PrintWriter(System.out);        long s = System.currentTimeMillis();        solve();        out.flush();        if (!oj){            System.out.println("[" + (System.currentTimeMillis() - s) + "ms]");        }    }    public boolean more(){        return in.hasNext();    }    public int ni(){        return in.nextInt();    }    public long nl(){        return in.nextLong();    }    public double nd(){        return in.nextDouble();    }    public String ns(){        return in.nextString();    }    public char nc(){        return in.nextChar();    }    class FastScanner {        BufferedReader br;        StringTokenizer st;        boolean hasNext;        public FastScanner(InputStream is) throws IOException {            br = new BufferedReader(new InputStreamReader(is));            hasNext = true;        }        public String nextToken() {            while (st == null || !st.hasMoreTokens()) {                try {                    st = new StringTokenizer(br.readLine());                } catch (Exception e) {                    hasNext = false;                    return "##";                }            }            return st.nextToken();        }        String next = null;        public boolean hasNext(){            next = nextToken();            return hasNext;        }        public int nextInt() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Integer.parseInt(more);        }        public long nextLong() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Long.parseLong(more);        }        public double nextDouble() {            if (next == null){                hasNext();            }            String more = next;            next = null;            return Double.parseDouble(more);        }        public String nextString(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more;        }        public char nextChar(){            if (next == null){                hasNext();            }            String more = next;            next = null;            return more.charAt(0);        }    }}