Problem: Btoh yuo adn yuor roomatme lhoate wianshg disehs, btu stlil sdmoeboy msut peorrfm tihs cohre dialy. Oen dya yuo decdie to idourtcne smoe syestm. Yuor rmmotaoe sstgegus teh fooniwllg dael. Yuo argee on tow arayrs of ientgres M adn R, nmebur upmicnog dyas (induiclng teh cunrret oen) wtih sicsescuve irnegets (teh ceurrnt dya is zreo), adn yuo wsah teh diehss on dya D if adn olny if terhe etsixs an iednx i scuh taht D mod M[i] = R[i], otwsehrie yuor rmootmae deos it. Yuo lkie teh cncepot, btu yuor rmotaome's cuinnng simle meaks yuo ssecupt sthnoemig, so yuo itennd to vefriy teh fnerisas of teh aemnrgeet.

Yuo aer geivn ayarrs M adn R. Cuaclatle teh pceanregte of dyas on wchih yuo edn up dnoig teh wisahng. Amsuse taht yuo hvae iiiftlneny mnay dyas aehad of yuo. 

Input

The first line of input contains a single integer N (1 ≤ N ≤ 16).

The second and third lines of input contain N integers each, all between 0 and 16, inclusive, and represent arrays M and R, respectively. All M[i] are positive, for each i R[i] < M[i].

Output

Output a single real number. The answer is considered to be correct if its absolute or relative error does not exceed 10 - 4.

Examples

Input

1
2
0


Output

0.500000


Input

2
2 3
1 0


Output

0.666667

Solution (in JAVA):

Solution:
import java.io.BufferedWriter;
import java.io.FileInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStreamWriter;
import java.io.PrintWriter;
import java.io.ObjectInputStream.GetField;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.InputMismatchException;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.Random;
import java.util.Scanner;
import java.util.Stack;
import java.util.TreeMap;
import java.util.TreeSet;

 
		public	class Q2 {
				
				static long MOD = 1000000007;
				static boolean b1,b[],bc[];
				static boolean boo[][][];
				static int ans1 = -1,ans2 = -1, root;
				static ArrayList<Integer>[] amp,amp1;
				static ArrayList<Pair>[] pmp;
				static int sum[],dist[],cnt[],ans[],size[],col[][];
				static int p = 0,cnt1 =0 ,cnt2 = 0;
				//static ArrayList<Pair3>[][][] pmp;
				static FasterScanner sc = new FasterScanner(System.in);
				static Queue<Integer> q = new LinkedList<>();
				static int arr[],start[],end[],color[],parent[];
				static PriorityQueue<Integer> pq;
				//static ArrayList<Integer> parent = new ArrayList<>();
				static LinkedList<Integer> fa;
				static BufferedWriter log;
				static TreeMap<Integer,Integer> tm = new TreeMap<>();
			//	static HashMap<Integer,Pair> hs;
				static Stack<Integer> stack = new Stack<>();
				static Pair prr[];
				static int dp[][];
				static long parent1[],parent2[],size1[],size2[],value[];
				static long Ans;
				public static void main(String[] args) throws Exception {
					/*new Thread(null, new Runnable() {
						public void run() {
							try {
								soln();
							} catch (Exception e) {
								System.out.println(e);
							}
						}
					}, "1", 1 << 26).start();*/
					soln();
			    }
				static int MAX = 1000005;
				static int n , m ,k, max, x1, x2;
				static HashSet<Pair> hs;
				static double DP[][][];
				public static void soln() throws IOException {
					//FasterScanner in = new FasterScanner(new FileInputStream("C:\\Users\\Admin\\Desktop\\QAL.txt"));
					//PrintWriter out = new PrintWriter("C:\\Users\\Admin\\Desktop\\A1.txt");
					log = new BufferedWriter(new OutputStreamWriter(System.out));
					int n = sc.nextInt();
					int m[] = sc.nextIntArray(n);
					int r[] = sc.nextIntArray(n);
					boolean b[] = new boolean[1000000];
					long lcm = m[0], gcd = m[0], mul =m[0];
					for(int i =1 ;i <n; i++){
						lcm = ((lcm*m[i])/gcd(lcm,m[i]));
					}
					//System.out.println(mul+" "+lcm);
					int cnt = 0;
					for(int i = 0; i<n ; i++){
						for(int j = 1;j<=lcm;j++){
							if(j%m[i]==r[i] && !b[j]){
								b[j] = true;
								cnt++;
							}
						}
					}
					//System.out.println(cnt);
					System.out.printf("%.9f",(double)cnt/lcm);
					log.close();
					//out.close();
				}
				 static long maxSubArraySum(long a[])
				    {
				        int size = a.length;
				        long max_so_far = Long.MIN_VALUE/2, max_ending_here = 0;

				        for (int i = 0; i < size; i++)
				        {
				            max_ending_here = max_ending_here + a[i];
				            if(max_so_far < max_ending_here)
				                max_so_far = max_ending_here;
				            if(max_ending_here < 0)
				                max_ending_here = 0;
				        }
				        return max_so_far;
				    }
				public class TrieNode{
					TrieNode[] t = new TrieNode[2];
					int cnt = 0;
				}
				public class Trie{
					TrieNode root;
					Trie(){
						root = new TrieNode();
					}
					public void insert(String s){
						TrieNode temp = root;
						for(int i = 0; i< s.length();i++){
							if(temp.t[s.charAt(i)-'0']==null){
								TrieNode n = new TrieNode();
								temp.t[s.charAt(i)-'0'] = n;
							}
							temp = temp.t[s.charAt(i)-'0'];
						}
						temp.cnt++;
					}
					public void delete(String s, TrieNode temp){
						
					}
					public String get(String s){
						StringBuilder sb = new StringBuilder();
						TrieNode temp = root;
						for(int i = 0; i< s.length();i++){
							int x = (s.charAt(i)-'0'+1)%2;
							if(temp.t[x]!=null){
								temp = temp.t[x];
								sb.append(1);
							}
							else{
								temp = temp.t[(x+1)%2];
								sb.append(0);
							}
						}
						return sb.toString();
					}
				}
				public static class FenwickTree {
				    int[] array; // 1-indexed array, In this array We save cumulative information to perform efficient range queries and updates
				    public FenwickTree(int size) {
				        array = new int[size + 1];
				    }
				    public int rsq(int ind) {
				        assert ind > 0;
				        int sum = 0;
				        while (ind > 0) {
				            sum += array[ind];ind -= ind & (-ind);
				        }
				        return sum;
				    }
				    public int rsq(int a, int b) {
				        assert b >= a && a > 0 && b > 0;
				        return rsq(b) - rsq(a - 1);
				    }
				    public void update(int ind, int value) {
				        assert ind > 0;
				        while (ind < array.length) {
				            array[ind] += value; ind += ind & (-ind);
				        }
				    }
				    public int size() {
				        return array.length - 1;
				    }
				}
				public static void dfs(int x){
					b[x] = true;
					for(int e:amp[x]){
						if(!b[e]){
							dfs(e);
						}
					}
				}
				public static void bfs(int x){
					b[x] = true;
					q = new LinkedList<Integer>();
					q.add(x);
					while(!q.isEmpty()){
						//System.out.println(q);
						int y = q.poll();
							for(int i:amp[y]){
							if(!b[i]){
								q.add(i);
								dist[i] = Math.max(dist[i], dist[y]+1);
								if(dist[i]>max){
									max = dist[i];
									k = i;
								}
								b[i]= true;
							}
						}
					}
				}
				static class Pair implements Comparable<Pair> {
					int u;
					int v;
					public Pair(){
						u = 0;
						v = 0;
					}
					public Pair(int u, int v) {
						this.u = u;
						this.v = v;
					}
					public int hashCode() {
						int hu = (int) (u ^ (u >>> 32));
						int hv = (int) (v ^ (v >>> 32));
						return 31*hu + hv;
					}
			 
					public boolean equals(Object o) {
						Pair other = (Pair) o;
						return (u == other.u && v == other.v);
					}
			 
					public int compareTo(Pair other) {
						return Long.compare(u, other.u) != 0 ? (Long.compare(u, other.u)) : (Long.compare(v, other.v));
					}
					
					public String toString() {
						return "[u=" + u + ", v=" + v + "]";
					}
				}
				
				public static long gcd(long gc, long m){
					if(m!=0) return gcd(m,gc%m);
					else return gc;
				}
				
				static int CeilIndex(int A[], int l, int r, int key)
			    {
			        while (r - l > 1)
			        {
			            int m = l + (r - l)/2;
			            if (A[m]>=key)
			                r = m;
			            else
			                l = m;
			        }
			 
			        return r;
			    }
			/*	public static Pair recur(int N, int K, int C){
					if(K>k) return new Pair(Long.MAX_VALUE/2,-1);
					if(N>=n && k<K) return new Pair(Long.MAX_VALUE/2,-1);
					if(N>=n && k==K) return new Pair(0,k);
					if(N>=n) return new Pair(0,0);
					if(dp[N][C][K]!=null){
						//System.out.println(dp[N][C][K]);
						return dp[N][C][K];
					} 
					Pair ans = new Pair(Long.MAX_VALUE/2,-1);
					if(arr[N]>0){
						Pair p;
						if(arr[N]==C) p = recur(N+1,K,arr[N]);
						else p = recur(N+1,K+1,arr[N]);
						if((p.u)<ans.u){
							ans = p;
						}
						if(p.u==ans.u){
							if(p.v>ans.v){
								ans = p;
							}
						}
					}
					else{
					for(int i = 1 ;i <= m; i++){
						Pair p;
						if(i==C)
						p = recur(N+1,K,i);
						else p = recur(N+1,K+1,i);
						p.u+=col[N][i-1];
						if((p.u)<ans.u){
							ans = p;
						}
						if(p.u==ans.u){
							if(p.v>ans.v){
								ans = p;
							}
						}
					}
					//System.out.println(ans);
					}
					return dp[N][C][K] = ans;
				}*/
				public static void mergeSort(int[] arr2, int l ,int r){
					if((r-l)>=1){
						int mid = (l+r)/2;
						mergeSort(arr2,l,mid);
						mergeSort(arr2,mid+1,r);
						merge(arr2,l,r,mid);
						//System.out.println(Ans);
					}
				}
				public static void merge(int arr[], int l, int r, int mid){
					int n1 = (mid-l+1), n2 = (r-mid);
					int left[] = new int[n1];
					int right[] = new int[n2];
					for(int i =0 ;i<n1;i++) left[i] = arr[l+i];
					for(int i =0 ;i<n2;i++) right[i] = arr[mid+1+i];
					int i =0, j =0, k = l;
					while(i<n1 && j<n2){
						if(left[i]>right[j]){
							cnt[right[j]]+=(n1-i);
							arr[k++] = right[j++];
						}
						else{
							arr[k++] = left[i++];
						}
					}
					while(i<n1) arr[k++] = left[i++];
					while(j<n2) arr[k++] = right[j++];
				}
				public static void buildGraph(int n){
					for(int i =0;i<n;i++){
						int x = sc.nextInt(), y = sc.nextInt();
						amp[x].add(y);
						hs.add(new Pair(x,y));
						hs.add(new Pair(y,x));
					}
				}
				
				
 
				public static int getParent(int x){
					while(parent[x]!=x){
						parent[x] = parent[ parent[x]];
						x = parent[ x];
					}
					return x;
				}
				
				static int min(int a, int b, int c){
					if(a<b && a<c) return a;
					if(b<c) return b;
					return c;
				}
				private static void permutation(String prefix, String str) {
				    int n = str.length();
				    if (n == 0); //hs.add(prefix);
				    else {
				        for (int i = 0; i < n; i++)
				            permutation(prefix + str.charAt(i), str.substring(0, i) + str.substring(i+1, n));
				    }
				}
				static class Graph{
					int vertex;
					long weight;
				}
				public static void buildTree(int n){
					int arr[] = sc.nextIntArray(n);
					for(int i = 0;i<n;i++){
						int x = arr[i]-1;
						amp[i+1].add(x);
						amp[x].add(i+1);
					}
				}
				static class Node{
					long max = 0, pre = 0, suf = 0, sum = 0;
					Node(long max, long pre, long suf, long sum){
						this.max = max;
						this.pre = pre;
						this.suf = suf;
						this.sum = sum;
					}
					Node(){};
				}
				static class SegmentTree {
					Node st[];
				//	long lazy[];
					SegmentTree(int n) {
						int size = 4 * n;
						st = new Node[size];
						for(int i = 0; i< size;i++) st[i] = new Node();
					//	lazy = new long[size];
						build(0, n - 1, 1);
					}
					Node comb(Node l,  Node r){
						long sum = l.sum+r.sum;
						long pre = Math.max(l.pre, l.sum+r.pre);
						long suf = Math.max(r.suf, r.sum+l.suf);
						long max = max(l.max, r.max, l.suf+r.pre);
						return new Node(max, pre , suf, sum);
					}
					Node build(int ss, int se, int si) {
						if(ss==se){
							st[si] = new Node(arr[ss],arr[ss],arr[ss],arr[ss]);
							return st[si];
						}
						int mid = (ss+se)/2;
						Node a1 = build(ss, mid, si*2), a2 = build(mid+1, se, si*2+1);
						return st[si] = comb(a1,a2);
					}
 
					void update(int si, int ss, int se, int idx, int val) {
						if (ss == se) {
							arr[ss] = val;
							st[si] = new Node(arr[ss],arr[ss],arr[ss],arr[ss]);
						} 
						else {
							int mid = (ss + se) / 2;
							if(ss <= idx && idx <= mid)
					        {
					             update(2*si, ss, mid, idx, val);
					        }
					        else
					        { update(2*si+1, mid+1, se, idx, val);
					        }
							st[si] = comb(st[si*2],st[si*2+1]);
						}
					}
					Node get(int qs, int qe, int ss, int se, int si){
						if(qs>se || qe<ss){
							return new Node(-10000000,-100000000,-10000000,-100000000);
						}
						if (qs <= ss && qe >= se) {
							return st[si];
						}
						int mid = (ss+se)/2;
						Node temp1 = get(qs, qe, ss, mid, si * 2), temp2 = get(qs, qe, mid + 1, se, si * 2 + 1);
						return st[si] = comb(temp1,temp2);
					}
					/*void updateRange(int node, int start, int end, int l, int r, long val)
					{
					    if(lazy[node] != 0)
					    { 
					        // This node needs to be updated
					        st[node] += (end - start + 1) * lazy[node];    // Update it
					        if(start != end)
					        {
					            lazy[node*2] += lazy[node];                  // Mark child as lazy
					            lazy[node*2+1] += lazy[node];                // Mark child as lazy
					        }
					        lazy[node] = 0;                                  // Reset it
					    }
					    if(start > end || start > r || end < l)              // Current segment is not within range [l, r]
					        return;
					    if(start >= l && end <= r)
					    {
					        // Segment is fully within range
					        st[node] += (end - start + 1) * val;
					        if(start != end)
					        {
					            // Not leaf node
					            lazy[node*2] += val;
					            lazy[node*2+1] += val;
					        }
					        return;
					    }
					    int mid = (start + end) / 2;
					    updateRange(node*2, start, mid, l, r, val);        // Updating left child
					    updateRange(node*2 + 1, mid + 1, end, l, r, val);   // Updating right child
					    st[node] = st[node*2] + st[node*2+1];        // Updating root with max value 
					}
 
					long queryRange(int node, int start, int end, int l, int r)
					{
					    if(start > end || start > r || end < l)
					        return 0L;         // Out of range
					    if(lazy[node] != 0)
					    {
					        // This node needs to be updated
					        st[node] += (end - start + 1) * lazy[node];            // Update it
					        if(start != end)
					        {
					            lazy[node*2] += lazy[node];         // Mark child as lazy
					            lazy[node*2+1] += lazy[node];    // Mark child as lazy
					        }
					        lazy[node] = 0L;                 // Reset it
					    }
					    if(start >= l && end <= r)             // Current segment is totally within range [l, r]
					        return  st[node];
					    int mid = (start + end) / 2;
					    long p1 = queryRange(node*2, start, mid, l, r);         // Query left child
					    long p2 = queryRange(node*2 + 1, mid + 1, end, l, r); // Query right child
					    return (p1 + p2);
					}*/
					void print() {
						for (int i = 0; i < st.length; i++) {
							//System.out.print(st[i].u+" "+st[i].v+" "+st[i].z+"\n");
						}
						System.out.println();
					}
				}
				
				static class Tree {
					ArrayList<Integer>[] g ;
					int T[]; // parent
					int L[]; // level
					int P[][]; // P[i][j] = 2^j th parent of i
 
					public Tree(int n) {
						T = new int[n];
						L = new int[n];
						P = new int[n][log2(n) + 1];
						g = (ArrayList<Integer>[])new ArrayList[n];
						L[0] = 0;
						T[0] = -1;
						for (int i = 0; i < n; i++) {
							g[i] = new ArrayList<>();
						}
						for (int i = 0; i < n ; i++) {
							int a = sc.nextInt();
							for(int j = 0; j<a; j++){
								int b = sc.nextInt()-1;
								g[i].add(b);
								g[b].add(i);
							}
						}
						dfs(0, -1);
						for (int j = 0; (1 << j) <= n; j++)
							pre(0, j,-1);
					}
 
					public int log2(int n) {
						return (int) (Math.log10(n) / Math.log10(2));
					}
 
					public void dfs(int node, int p) {
						for (int k : g[node]) {
							if (k != p) {
								T[k] = node;
								L[k] = L[node] + 1;
								dfs(k, node);
							}
						}
					}
					public void pre(int node, int j,int p) {
						if (j == 0) {
							P[node][j] = T[node];
						} else {
							P[node][j] = P[P[node][j - 1]][j - 1];
						}
						for (int k : g[node]) {
							if(k!=p)
								pre(k, j,node);
						}
					}
					public int lca(int u, int v) {
						if (L[u] > L[v]) {
							int temp = u;
							u = v;
							v = temp;
						}
						int log;
						for (log = 0; (1 << log) <= L[v]; log++)
							; // convert level into log
						log--;
 
						for (int i = log; i >= 0; i--) {
							if ((1 << i) <= L[v] - L[u]) // make both node's level same
								v = P[v][i];
						}
						if (u == v)
							return u;
						for (int i = log; i >= 0; i--) {
							if (P[u][i] != -1 && P[u][i] != P[v][i]) // shift up both u and v
							{
								u = P[u][i];
								v = P[v][i];
							}
						}
						return T[u];
					}
				}
				public static long max(long x, long y, long z){
					if(x>=y && x>=z) return x;
					if(y>=x && y>=z) return y;
					return z;
				}
				
				public static void seive(long n){
					b = new boolean[(int) (n+1)];
					Arrays.fill(b, true);
					for(int i = 2;i*i<=n;i++){
						if(b[i]){
							for(int p = 2*i;p<=n;p+=i){
								b[p] = false;
							}
						}
					}
				}
				static long modInverse(long a, long mOD2){
				            return  power(a, mOD2-2, mOD2);
				}
				static long power(long x, long y, long m)
				{
				    if (y == 0)
				        return 1;
				    long p = power(x, y/2, m) % m;
				    p = (p * p) % m;
				 
				    return (y%2 == 0)? p : (x * p) % m;
				}
				static long d,x,y;
				public static void extendedEuclidian(long a, long b){
					if(b == 0) {
				        d = a;
				        x = 1;
				        y = 0;
				    }
				    else {
				        extendedEuclidian(b, a%b);
				        int temp = (int) x;
				        x = y;
				        y = temp - (a/b)*y;
				    }
				}
				
				
				static class FasterScanner {
 
					private final InputStream stream;
					private final byte[] buf = new byte[8192];
					private int curChar, snumChars;
					private SpaceCharFilter filter;
					public int read() {
						if (snumChars == -1)
							throw new InputMismatchException();
						if (curChar >= snumChars) {
							curChar = 0;
							try {
								snumChars = System.in.read(buf);
							} catch (IOException e) {
								throw new InputMismatchException();
							}
							if (snumChars <= 0)
								return -1;
						}
						return buf[curChar++];
					}
					public FasterScanner(InputStream stream) {
						this.stream = stream;
					}
					public String nextString() {
						int c = read();
						while (isSpaceChar(c))
							c = read();
						StringBuilder res = new StringBuilder();
						do {
							res.appendCodePoint(c);
							c = read();
						} while (!isSpaceChar(c));
						return res.toString();
					}
					public int snext() {
						if (snumChars == -1)
							throw new InputMismatchException();
						if (curChar >= snumChars) {
							curChar = 0;
							try {
								snumChars = stream.read(buf);
							} catch (IOException e) {
								throw new InputMismatchException();
							}
							if (snumChars <= 0)
								return -1;
						}
						return buf[curChar++];
					}
 
					public int nextInt() {
						int c = snext();
						while (isSpaceChar(c)) {
							c = snext();
						}
						int sgn = 1;
						if (c == '-') {
							sgn = -1;
							c = snext();
						}
						int res = 0;
						do {
							if (c < '0' || c > '9')
								throw new InputMismatchException();
							res *= 10;
							res += c - '0';
							c = snext();
						} while (!isSpaceChar(c));
						return res * sgn;
					}
 
					public long nextLong() {
						int c = snext();
						while (isSpaceChar(c)) {
							c = snext();
						}
						int sgn = 1;
						if (c == '-') {
							sgn = -1;
							c = snext();
						}
						long res = 0;
						do {
							if (c < '0' || c > '9')
								throw new InputMismatchException();
							res *= 10;
							res += c - '0';
							c = snext();
						} while (!isSpaceChar(c));
						return res * sgn;
					}
 
					public int[] nextIntArray(int n) {
						int a[] = new int[n];
						for (int i = 0; i < n; i++) {
							a[i] = nextInt();
						}
						return a;
					}
 
					public long[] nextLongArray(int n) {
						long a[] = new long[n];
						for (int i = 0; i < n; i++) {
							a[i] = nextInt();
						}
						return a;
					}
			                
					public String readString() {
						int c = snext();
						while (isSpaceChar(c)) {
							c = snext();
						}
						StringBuilder res = new StringBuilder();
						do {
							res.appendCodePoint(c);
							c = snext();
						} while (!isSpaceChar(c));
						return res.toString();
					}
 
					public String nextLine() {
						int c = snext();
						while (isSpaceChar(c))
							c = snext();
						StringBuilder res = new StringBuilder();
						do {
							res.appendCodePoint(c);
							c = snext();
						} while (!isEndOfLine(c));
						return res.toString();
					}
 
					public boolean isSpaceChar(int c) {
						if (filter != null)
							return filter.isSpaceChar(c);
						return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
					}
 
					private boolean isEndOfLine(int c) {
						return c == '\n' || c == '\r' || c == -1;
					}
 
					public interface SpaceCharFilter {
						public boolean isSpaceChar(int ch);
					}
				}
			}