using System; using System.IO; using System.Collections.Generic; using System.Linq; using System.Numerics; using System.Diagnostics; using System.Runtime.CompilerServices; namespace FertiLib.Contest.E { static class Program { public static void Solve(Scanner cin) { int n = cin.ReadInt(); var a = cin.ReadIntArray(n); var q = cin.ReadInt(); var prime = new List(); PrimeEnumerator.EnumPrime(2000, prime); var primedic = new SortedDictionary(); for (int i = 0; i < prime.Count; i++) { primedic.Add(prime[i], i); } var seg = new SegmentTree[prime.Count]; for (int i = 0; i < prime.Count; i++) { var div = new int[n]; for (int j = 0; j < n; j++) { int t = 0; int tmp = a[j]; while (tmp % prime[i] == 0) { t++; tmp++; } div[j] = t; } seg[i] = new SegmentTree(div, (x, y) => x + y, 0); } for (int i = 0; i < q; i++) { var (p, l, r) = cin.ReadValue(); bool ok = true; var factor = Factorizer.Factorize(p); foreach (var e in factor) { if (e.Key > 2000) { ok = false; break; } if (seg[primedic[(int)e.Key]].GetResult(l - 1, r) < e.Value) { ok = false; break; } } Console.WriteLine(ok ? "Yes" : "NO"); } } public static void Main(string[] args) { var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; Console.SetOut(sw); var cin = new Scanner(); Solve(cin); Console.Out.Flush(); } public static void YESNO(bool condition) => Console.WriteLine(condition ? "YES" : "NO"); public static void YesNo(bool condition) => Console.WriteLine(condition ? "Yes" : "No"); public static void yesno(bool condition) => Console.WriteLine(condition ? "yes" : "no"); public static bool Chmax(ref T a, T b) where T : IComparable { if (a.CompareTo(b) >= 0) return false; a = b; return true; } public static bool Chmin(ref T a, T b) where T : IComparable { if (a.CompareTo(b) <= 0) return false; a = b; return true; } } public static class Factorizer { // http://miller-rabin.appspot.com/ static readonly long[] baseSingle = { 126401071349994536 }; // <= 291,831 static readonly long[] baseDouble = { 336781006125, 9639812373923155 }; // <= 1,050,535,501 static readonly long[] baseQuad = { 2, 2570940, 211991001, 3749873356 }; // <= 47,636,622,961,201 static readonly long[] baseBig = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022 }; /// /// Factorizes n. /// /// The number to factorize. /// Dictionary that has prime factors in Key, the number of each factor in Value. public static SortedDictionary Factorize(long n) { var ret = new SortedDictionary(); var que = new Queue(); que.Enqueue(n); while (que.Count > 0) { var now = que.Dequeue(); if (now == 1) continue; if (IsPrime(now)) { if (ret.ContainsKey(now)) ret[now]++; else ret.Add(now, 1); continue; } long f = FindFactor(now); que.Enqueue(f); que.Enqueue(now / f); } return ret; } /// /// Tests if n is prime or not using Miller-Rabin Algorithm. Complexity: O(log^2 n) /// /// The number to test. /// True if prime, False otherwise. public static bool IsPrime(long n) { if (n == 1) return false; if (n == 2) return true; long d = n - 1; int s = 0; while (d % 2 == 0) { d /= 2; s++; } long[] bases; if (n < 291831) bases = baseSingle; else if (n < 1050535501) bases = baseDouble; else if (n < 47636622961201) bases = baseQuad; else bases = baseBig; foreach (var e in bases) { if (!MillerRabinTest(e, d, n, s)) return false; } return true; } private static bool MillerRabinTest(long e, long d, long n, long s) { long pow = ModPow(e, d, n); if (pow == 1) return true; for (int i = 0; i < s; i++) { if (pow == n - 1) return true; pow = ModPow(pow, 2, n); } return false; } private static long ModPow(long a, long p, long mod) { if (mod <= int.MaxValue) { a %= mod; long ret = 1; for (long i = 1; i <= p; i *= 2) { if (p / i % 2 == 1) { ret *= a; ret %= mod; } a *= a; a %= mod; } return ret; } else { BigInteger retbig = 1; BigInteger abig = a % mod; for (BigInteger i = 1; i <= p; i *= 2) { if (p / i % 2 == 1) { retbig *= abig; retbig %= mod; } abig *= abig; abig %= mod; } return (long)retbig; } } /// /// Finds a factor of n by Pollard's ρ algorithm. Complexity: O(n^(1/4)) /// /// The number to get a factor. /// A prime factor of n. public static long FindFactor(long n) { if (n % 2 == 0) return 2; long i = 0; while (true) { i++; long x = i; long y = Next(i, n); while (true) { long g = GCD(x - y, n); if (g >= n) break; if (1 < g) return g; x = Next(x, n); y = Next(y, n); y = Next(y, n); } } } private static long Next(long x, long mod) { BigInteger tmp = x; tmp *= tmp; tmp++; tmp %= mod; return (long)tmp; } private static long GCD(long a, long b) { a = Math.Abs(a); b = Math.Abs(b); while (a > 0) { b %= a; var x = a; a = b; b = x; } return b; } } public static class PrimeEnumerator { public static bool[] EnumPrime(int max, List prime = null) { var ret = new bool[max + 1]; for (int i = 2; i < max + 1; i++) { ret[i] = true; } for (int i = 2; i * i <= max; i++) { if (!ret[i]) continue; for (int j = i * i; j <= max; j += i) { ret[j] = false; } } if (prime != null) { for (int i = 2; i < max + 1; i++) { if (ret[i]) prime.Add(i); } } return ret; } } public class SegmentTree { public int Count { get; private set; } T[] tree; readonly Func func; readonly T init; public SegmentTree(int size, Func func, T init) { int n = 1; while (n <= size) { n *= 2; } Count = n; tree = Enumerable.Repeat(init, 2 * n - 1).ToArray(); this.func = func; this.init = init; } public SegmentTree(IEnumerable items, Func func, T init) : this(items.Count(), func, init) { var array = items.ToArray(); for (int i = 0; i < array.Length; i++) { tree[i + Count - 1] = array[i]; } UpdateAll(); } public T this[int index] { get => this.tree[index + Count - 1]; private set => this.tree[index + Count - 1] = value; } public void Update(int index, T value) { index += Count - 1; tree[index] = value; while (index > 0) { index = (index - 1) / 2; tree[index] = func(tree[index * 2 + 1], tree[index * 2 + 2]); } } void UpdateAll() { int n = Count / 2; while (n > 0) { for (int i = 0; i < n; i++) { tree[n - 1 + i] = func(tree[2 * (n - 1 + i) + 1], tree[2 * (n - 1 + i) + 2]); } n /= 2; } } // Attention: [l, r). This result does not include r-th item. public T GetResult(int l, int r) => GetResult(l, r, 0, 0, Count); T GetResult(int l, int r, int k, int a, int b) { if (b <= l || r <= a) return init; if (l <= a && b <= r) return tree[k]; var vl = GetResult(l, r, 2 * k + 1, a, (a + b) / 2); var vr = GetResult(l, r, 2 * k + 2, (a + b) / 2, b); return func(vl, vr); } } static class Extention { public static string Join(this IEnumerable x, string separator = "") => string.Join(separator, x); public static int UpperBound(this IList list, T value) => list.BinarySearch(value, true, 0, list.Count, Comparer.Default); public static int LowerBound(this IList list, T value) => list.BinarySearch(value, false, 0, list.Count, Comparer.Default); public static int BinarySearch(this IList list, T value, bool isUpperBound, int index, int length, Comparer comparer) { var ng = index - 1; var ok = index + length; while (ok - ng > 1) { var mid = ng + (ok - ng) / 2; var res = comparer.Compare(list[mid], value); if (res < 0 || (isUpperBound && res == 0)) ng = mid; else ok = mid; } return ok; } } class Scanner { string[] s; int i; char[] separator = new char[] { ' ' }; public Scanner() { s = new string[0]; i = 0; } public string Read() => ReadString(); public string ReadString() { if (i < s.Length) return s[i++]; string st = Console.ReadLine(); while (st == "") st = Console.ReadLine(); s = st.Split(separator, StringSplitOptions.RemoveEmptyEntries); if (s.Length == 0) return ReadString(); i = 0; return s[i++]; } public string[] ReadStringArray(int N) { string[] Array = new string[N]; for (int i = 0; i < N; i++) { Array[i] = ReadString(); } return Array; } public int ReadInt() { return int.Parse(ReadString()); } public int[] ReadIntArray(int N, int add = 0) { int[] Array = new int[N]; for (int i = 0; i < N; i++) { Array[i] = ReadInt() + add; } return Array; } public long ReadLong() { return long.Parse(ReadString()); } public long[] ReadLongArray(int N, long add = 0) { long[] Array = new long[N]; for (int i = 0; i < N; i++) { Array[i] = ReadLong() + add; } return Array; } public double ReadDouble() { return double.Parse(ReadString()); } public double[] ReadDoubleArray(int N, double add = 0) { double[] Array = new double[N]; for (int i = 0; i < N; i++) { Array[i] = ReadDouble() + add; } return Array; } public T1 ReadValue() => (T1)Convert.ChangeType(ReadString(), typeof(T1)); public (T1, T2) ReadValue() { var inputs = ReadStringArray(2); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); return (v1, v2); } public (T1, T2, T3) ReadValue() { var inputs = ReadStringArray(3); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3)); return (v1, v2, v3); } public (T1, T2, T3, T4) ReadValue() { var inputs = ReadStringArray(4); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3)); var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4)); return (v1, v2, v3, v4); } public (T1, T2, T3, T4, T5) ReadValue() { var inputs = ReadStringArray(5); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3)); var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4)); var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5)); return (v1, v2, v3, v4, v5); } public (T1, T2, T3, T4, T5, T6) ReadValue() { var inputs = ReadStringArray(6); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3)); var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4)); var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5)); var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6)); return (v1, v2, v3, v4, v5, v6); } public (T1, T2, T3, T4, T5, T6, T7) ReadValue() { var inputs = ReadStringArray(7); var v1 = (T1)Convert.ChangeType(inputs[0], typeof(T1)); var v2 = (T2)Convert.ChangeType(inputs[1], typeof(T2)); var v3 = (T3)Convert.ChangeType(inputs[2], typeof(T3)); var v4 = (T4)Convert.ChangeType(inputs[3], typeof(T4)); var v5 = (T5)Convert.ChangeType(inputs[4], typeof(T5)); var v6 = (T6)Convert.ChangeType(inputs[5], typeof(T6)); var v7 = (T7)Convert.ChangeType(inputs[6], typeof(T7)); return (v1, v2, v3, v4, v5, v6, v7); } public T1[] ReadValueArray(int N) { var v1 = new T1[N]; for (int i = 0; i < N; i++) { v1[i] = ReadValue(); } return v1; } public (T1[], T2[]) ReadValueArray(int N) { var (v1, v2) = (new T1[N], new T2[N]); for (int i = 0; i < N; i++) { var (t1, t2) = ReadValue(); v1[i] = t1; v2[i] = t2; } return (v1, v2); } public (T1[], T2[], T3[]) ReadValueArray(int N) { var (v1, v2, v3) = (new T1[N], new T2[N], new T3[N]); for (int i = 0; i < N; i++) { var (t1, t2, t3) = ReadValue(); v1[i] = t1; v2[i] = t2; v3[i] = t3; } return (v1, v2, v3); } public (T1[], T2[], T3[], T4[]) ReadValueArray(int N) { var (v1, v2, v3, v4) = (new T1[N], new T2[N], new T3[N], new T4[N]); for (int i = 0; i < N; i++) { var (t1, t2, t3, t4) = ReadValue(); v1[i] = t1; v2[i] = t2; v3[i] = t3; v4[i] = t4; } return (v1, v2, v3, v4); } public (T1[], T2[], T3[], T4[], T5[]) ReadValueArray(int N) { var (v1, v2, v3, v4, v5) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N]); for (int i = 0; i < N; i++) { var (t1, t2, t3, t4, t5) = ReadValue(); v1[i] = t1; v2[i] = t2; v3[i] = t3; v4[i] = t4; v5[i] = t5; } return (v1, v2, v3, v4, v5); } public (T1[], T2[], T3[], T4[], T5[], T6[]) ReadValueArray(int N) { var (v1, v2, v3, v4, v5, v6) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N], new T6[N]); for (int i = 0; i < N; i++) { var (t1, t2, t3, t4, t5, t6) = ReadValue(); v1[i] = t1; v2[i] = t2; v3[i] = t3; v4[i] = t4; v5[i] = t5; v6[i] = t6; } return (v1, v2, v3, v4, v5, v6); } public (T1[], T2[], T3[], T4[], T5[], T6[], T7[]) ReadValueArray(int N) { var (v1, v2, v3, v4, v5, v6, v7) = (new T1[N], new T2[N], new T3[N], new T4[N], new T5[N], new T6[N], new T7[N]); for (int i = 0; i < N; i++) { var (t1, t2, t3, t4, t5, t6, t7) = ReadValue(); v1[i] = t1; v2[i] = t2; v3[i] = t3; v4[i] = t4; v5[i] = t5; v6[i] = t6; v7[i] = t7; } return (v1, v2, v3, v4, v5, v6, v7); } } }