Problem:
Score : 300 points

Problem Statement

You are given a string S of length N and another string T of length M.
These strings consist of lowercase English letters.

A string X is called a good string when the following conditions are all met:

Let L be the length of X. L is divisible by both N and M.

Concatenating the 1-st, (\frac{L}{N}+1)-th, (2 \times \frac{L}{N}+1)-th, ..., ((N-1)\times\frac{L}{N}+1)-th characters of X, without changing the order, results in S.

Concatenating the 1-st, (\frac{L}{M}+1)-th, (2 \times \frac{L}{M}+1)-th, ..., ((M-1)\times\frac{L}{M}+1)-th characters of X, without changing the order, results in T.

Determine if there exists a good string. If it exists, find the length of the shortest such string.

Constraints

1 \leq N,M \leq 10^5

S and T consist of lowercase English letters.

|S|=N

|T|=M

Input

Input is given from Standard Input in the following format:

N M
S
T

Output

If a good string does not exist, print -1; if it exists, print the length of the shortest such string.

Sample Input 1

3 2
acp
ae

Sample Output 1

6

For example, the string accept is a good string.
There is no good string shorter than this, so the answer is 6.

Sample Input 2

6 3
abcdef
abc

Sample Output 2

-1

Sample Input 3

15 9
dnsusrayukuaiia
dujrunuma

Sample Output 3

45

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
//using System.Numerics;  //comment out if AOJ
using System.Text;

using Problem = Tmp.Problem;
using MyIO;

#pragma warning disable   //for AOJ

namespace Tmp
{
    class Problem : IDisposable
    {
        bool IsGCJ;
        int Repeat;
        Scanner sc;
        Printer pr;
        public Problem(bool isGCJ, Scanner scanner, Printer printer)
        {
            sc = scanner;
            pr = printer;
            IsGCJ = isGCJ;
            if (isGCJ) Repeat = sc.nextInt();
            else Read();
        }
        public Problem(bool isGCJ) : this(isGCJ, new Scanner(), new Printer()) { }
        public Problem(bool isGCJ, Scanner scanner) : this(isGCJ, scanner, new Printer()) { }
        public Problem(bool isGCJ, Printer printer) : this(isGCJ, new Scanner(), printer) { }
        public void Solve()
        {
            if (IsGCJ) for (var i = 0; i < Repeat; i++) { Read(); pr.Write("Case #" + (i + 1) + ": "); SolveOne(); }
            else SolveOne();
        }
        public void Dispose()
        {
            sc.Dispose();
            pr.Dispose();
        }
        public int Size { get { return 1; } }
        public const long Mod = 1000000007;

        // 使用する変数をここに書く
        // string S;
        // int a;
        /// <summary>
        /// 読み込み処理をここに書く
        /// </summary>
        void Read()
        {

        }
        /// <summary>
        /// メイン処理をここに書く
        /// </summary>
        void SolveOne()
        {
            int n = sc.nextInt();
            int m = sc.nextInt();
            string s = sc.next();
            string t = sc.next();

            long l = Func.LCM(n, m);
            bool ok = true;
            int scnt = 0;
            int tcnt = 0;
            for(; ;)
            {
                if(scnt * l/n == tcnt * l /m)
                {
                   if( s[scnt] ==t[tcnt])
                    {
                        scnt++;
                        tcnt++;
                    }
                   else
                    {
                        ok = false;
                        break;
                    }
                }
                else if (scnt * l / n > tcnt * l / m)
                {
                    tcnt++;
                }
                else
                {
                    scnt++;
                }

                if(scnt >= n || tcnt >= m)
                {
                    break;
                }
            }

            if (ok == false) l = -1;
            Console.WriteLine(l);
        }
    }
}
class Program
{
    //public static RandomSFMT rand = new RandomSFMT();
    public static bool IsJudgeMode = true;
    public static bool IsGCJMode = false;
    public static bool IsSolveCreated = true;
    static void Main()
    {
        if (IsJudgeMode)
            if (IsGCJMode) using (var problem = new Problem(true, new Scanner("C-large-practice.in.txt"), new Printer("output.txt"))) problem.Solve();
            else using (var problem = new Problem(false, new Printer())) problem.Solve();
        else
        {
            var num = 1;
            int size = 0;
            decimal time = 0;
            for (var tmp = 0; tmp < num; tmp++)
            {
                using (var P = IsSolveCreated ? new Problem(false, new Scanner("input.txt"), new Printer()) : new Problem(false))
                {
                    size = P.Size;
                    //time += Func.MeasureTime(() => P.Solve());
                }
            }
            Console.WriteLine("{0}, {1}ms", size, time / num);
        }
    }
}

/// <summary>
/// カスタマイズしたIO
/// </summary>
namespace MyIO
{
    class Printer : IDisposable
    {
        bool isConsole;
        TextWriter file;
        public Printer() { file = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false }; isConsole = true; }
        public Printer(string path) { file = new StreamWriter(path, false) { AutoFlush = false }; isConsole = false; }
        public void Write<T>(T value) { file.Write(value); }
        public void Write(bool b) { file.Write(b ? "YES" : "NO"); }
        public void Write(string str, params object[] args) { file.Write(str, args); }
        public void WriteLine() { file.WriteLine(); }
        public void WriteLine<T>(T value) { file.WriteLine(value); }
        public void WriteLine(bool b) { file.WriteLine(b ? "YES" : "NO"); }
        public void WriteLine<T>(IEnumerable<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(List<T> list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine<T>(T[] list) { foreach (var x in list) file.WriteLine(x); }
        public void WriteLine(string str, params object[] args) { file.WriteLine(str, args); }
        public void Dispose() { file.Flush(); if (!isConsole) file.Dispose(); }
    }
    class Scanner : IDisposable
    {
        bool isConsole;
        TextReader file;
        public Scanner() { file = Console.In; }
        public Scanner(string path) { file = new StreamReader(path); isConsole = false; }
        public void Dispose() { if (!isConsole) file.Dispose(); }

        #region next読み込み
        string[] nextBuffer = new string[0];
        int BufferCnt = 0;

        char[] cs = new char[] { ' ' };

        public string next()
        {
            while (BufferCnt >= nextBuffer.Length)
            {
                string st = file.ReadLine();
                while (st == "") st = file.ReadLine();
                nextBuffer = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
                BufferCnt = 0;
            }
            return nextBuffer[BufferCnt++];
        }

        public int nextInt()
        {
            return int.Parse(next());
        }

        public long nextLong()
        {
            return long.Parse(next());
        }

        public double nextDouble()
        {
            return double.Parse(next());
        }

        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public string[] next(int n) { return enumerate(n, next); }
        public double[] nextDouble(int n) { return enumerate(n, nextDouble); }
        public int[] nextInt(int n) { return enumerate(n, nextInt); }
        public long[] nextLong(int n) { return enumerate(n, nextLong); }
        #endregion
    }
}


/// <summary>
/// 共通機能
/// </summary>
static partial class Func
{
    public const int Inf = 1073741789;  // 2 * Inf < int.MaxValue, and Inf is a prime number
    public const long InfL = 4011686018427387913L;  // 2 * InfL < long.MaxValue, and InfL is a prime number
    public static Comparison<T> DefaultComparison<T>() { return (x, y) => Comparer<T>.Default.Compare(x, y); }
    public static Comparison<T> ToComparison<T>(this IComparer<T> comp) { return comp == null ? DefaultComparison<T>() : (x, y) => comp.Compare(x, y); }
    /// <summary>
    /// Find the first number x such that pred(x) is true
    /// if pred(x) is false for all min&lt;=x&lt;max, then return max
    /// in other words, pred(max) is assumed to be true
    /// </summary>
    /// <param name="min">inclusive lower limit</param>
    /// <param name="max">exclusive upper limit</param>
    /// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
    /// <returns>first number such that satisfy pred</returns>
    public static long FirstBinary(long min, long max, Predicate<long> pred)
    {
        while (min < max)
        {
            var mid = (min + max) / 2;
            if (pred(mid)) max = mid;
            else min = mid + 1;
        }
        return min;
    }
    /// <summary>
    /// Find the first number x such that pred(x) is true
    /// if pred(x) is false for all min&lt;=x&lt;max, then return max
    /// in other words, pred(max) is assumed to be true
    /// </summary>
    /// <param name="min">inclusive lower limit</param>
    /// <param name="max">exclusive upper limit</param>
    /// <param name="pred">monotonous predicate, i.e. if pred(a) and a&lt;b, then pred(b)</param>
    /// <returns>first number such that satisfy pred</returns>
    public static int FirstBinary(int min, int max, Predicate<int> pred)
    {
        while (min < max)
        {
            var mid = (min + max) / 2;
            if (pred(mid)) max = mid;
            else min = mid + 1;
        }
        return min;
    }
    public static void Swap<T>(this IList<T> array, int i, int j) { var tmp = array[i]; array[i] = array[j]; array[j] = tmp; }
    public static void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
}


/// <summary>
/// C++のSTLを再実装
/// </summary>
#region STL
class PriorityQueue<T> : IEnumerable<T>, ICollection, IEnumerable, ICloneable
{
    Comparison<T> comp;
    List<T> list;
    public int Count { get; private set; }
    public bool IsEmpty { get { return Count == 0; } }
    public PriorityQueue(IEnumerable<T> source) : this((Comparison<T>)null, 0, source) { }
    public PriorityQueue(int capacity = 0, IEnumerable<T> source = null) : this((Comparison<T>)null, capacity, source) { }
    public PriorityQueue(IComparer<T> comp, IEnumerable<T> source) : this(comp.ToComparison(), source) { }
    public PriorityQueue(IComparer<T> comp, int capacity = 0, IEnumerable<T> source = null) : this(comp.ToComparison(), source) { list.Capacity = capacity; }
    public PriorityQueue(Comparison<T> comp, IEnumerable<T> source) : this(comp, 0, source) { }
    public PriorityQueue(Comparison<T> comp, int capacity = 0, IEnumerable<T> source = null) { this.comp = comp == null ? Func.DefaultComparison<T>() : comp; list = new List<T>(capacity); if (source != null) foreach (var x in source) Enqueue(x); }
    /// <summary>
    /// add an item
    /// this is an O(log n) operation
    /// </summary>
    /// <param name="x">item</param>
    public void Enqueue(T x)
    {
        var pos = Count++;
        list.Add(x);
        while (pos > 0)
        {
            var p = (pos - 1) / 2;
            if (comp(list[p], x) <= 0) break;
            list[pos] = list[p];
            pos = p;
        }
        list[pos] = x;
    }
    /// <summary>
    /// return the minimum element and remove it
    /// this is an O(log n) operation
    /// </summary>
    /// <returns>the minimum</returns>
    public T Dequeue()
    {
        var value = list[0];
        var x = list[--Count];
        list.RemoveAt(Count);
        if (Count == 0) return value;
        var pos = 0;
        while (pos * 2 + 1 < Count)
        {
            var a = 2 * pos + 1;
            var b = 2 * pos + 2;
            if (b < Count && comp(list[b], list[a]) < 0) a = b;
            if (comp(list[a], x) >= 0) break;
            list[pos] = list[a];
            pos = a;
        }
        list[pos] = x;
        return value;
    }
    /// <summary>
    /// look at the minimum element
    /// this is an O(1) operation
    /// </summary>
    /// <returns>the minimum</returns>
    public T Peek() { return list[0]; }
    public IEnumerator<T> GetEnumerator() { var x = (PriorityQueue<T>)Clone(); while (x.Count > 0) yield return x.Dequeue(); }
    void CopyTo(Array array, int index) { foreach (var x in this) array.SetValue(x, index++); }
    public object Clone() { var x = new PriorityQueue<T>(comp, Count); x.list.AddRange(list); return x; }
    public void Clear() { list = new List<T>(); Count = 0; }
    public void TrimExcess() { list.TrimExcess(); }
    /// <summary>
    /// check whether item is in this queue
    /// this is an O(n) operation
    /// </summary>
    public bool Contains(T item) { return list.Contains(item); }
    IEnumerator IEnumerable.GetEnumerator() { return GetEnumerator(); }
    void ICollection.CopyTo(Array array, int index) { CopyTo(array, index); }
    bool ICollection.IsSynchronized { get { return false; } }
    object ICollection.SyncRoot { get { return this; } }
}
class Deque<T>
{
    T[] array;
    int offset, capacity;
    public int Count { get; protected set; }
    public Deque(int capacity) { array = new T[this.capacity = capacity]; Count = 0; offset = 0; }
    public Deque() : this(16) { }
    public T this[int index] { get { return array[GetIndex(index)]; } set { array[GetIndex(index)] = value; } }
    int GetIndex(int index) { var tmp = index + offset; return tmp >= capacity ? tmp - capacity : tmp; }
    public T PeekFront() { return array[offset]; }
    public T PeekBack() { return array[GetIndex(Count - 1)]; }
    public void PushFront(T item)
    {
        if (Count == capacity) Extend();
        if (--offset < 0) offset += array.Length;
        array[offset] = item;
        Count++;
    }
    public T PopFront()
    {
        Count--;
        var tmp = array[offset++];
        if (offset >= capacity) offset -= capacity;
        return tmp;
    }
    public void PushBack(T item)
    {
        if (Count == capacity) Extend();
        var id = (Count++) + offset;
        if (id >= capacity) id -= capacity;
        array[id] = item;
    }
    public T PopBack() { return array[GetIndex(--Count)]; }
    public void Insert(int index, T item)
    {
        PushFront(item);
        for (var i = 0; i < index; i++) this[i] = this[i + 1];
        this[index] = item;
    }
    public T RemoveAt(int index)
    {
        var tmp = this[index];
        for (var i = index; i > 0; i--) this[i] = this[i - 1];
        PopFront();
        return tmp;
    }
    void Extend()
    {
        var newArray = new T[capacity << 1];
        if (offset > capacity - Count)
        {
            var length = array.Length - offset;
            Array.Copy(array, offset, newArray, 0, length);
            Array.Copy(array, 0, newArray, length, Count - length);
        }
        else Array.Copy(array, offset, newArray, 0, Count);
        array = newArray;
        offset = 0;
        capacity <<= 1;
    }
}
class PairComparer<S, T> : IComparer<Pair<S, T>>
    where S : IComparable<S>
    where T : IComparable<T>
{
    public PairComparer() { }
    public int Compare(Pair<S, T> x, Pair<S, T> y)
    {
        var p = x.First.CompareTo(y.First);
        if (p != 0) return p;
        else return x.Second.CompareTo(y.Second);
    }
}
class Pair<S, T>
{
    public S First;
    public T Second;
    public Pair() { First = default(S); Second = default(T); }
    public Pair(S s, T t) { First = s; Second = t; }
    public override string ToString() { return string.Format("({0}, {1})", First, Second); }
    public override int GetHashCode() { return First.GetHashCode() ^ Second.GetHashCode(); }
    public override bool Equals(object obj)
    {
        if (ReferenceEquals(this, obj)) return true;
        else if (obj == null) return false;
        var tmp = obj as Pair<S, T>;
        return (object)tmp != null && First.Equals(tmp.First) && Second.Equals(tmp.Second);
    }
}
#endregion //STL



#region MyMath
static partial class Func
{
    #region BasicMath
    //TODO: 基本セットの整備(extgcdとか)

    /// <summary>
    /// 最大公約数
    /// </summary>
    /// <param name="n"></param>
    /// <param name="m"></param>
    /// <returns></returns>
    public static int GCD(int n, int m)
    {
        var a = Math.Abs(n);
        var b = Math.Abs(m);
        if (a < b) { var c = a; a = b; b = c; }
        while (b > 0)
        {
            var c = a % b;
            a = b;
            b = c;
        }
        return a;
    }
    /// <summary>
    /// 使わないで
    /// </summary>
    /// <param name="n"></param>
    /// <param name="m"></param>
    /// <returns></returns>
    public static long GCD_oldver(long n, long m)
    {
        var a = Math.Abs(n);
        var b = Math.Abs(m);
        if (a < b) { var c = a; a = b; b = c; }
        while (b > 0)
        {
            var c = a % b;
            a = b;
            b = c;
        }
        return a;
    }
    /// <summary>
    /// 最大公約数
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public static long GCD(long a, long b)
    {
        var n = (ulong)Math.Abs(a); var m = (ulong)Math.Abs(b);
        if (n == 0) return (long)m; if (m == 0) return (long)n;
        int zm = 0, zn = 0;
        while ((n & 1) == 0) { n >>= 1; zn++; }
        while ((m & 1) == 0) { m >>= 1; zm++; }
        while (m != n)
        {
            if (m > n) { m -= n; while ((m & 1) == 0) m >>= 1; }
            else { n -= m; while ((n & 1) == 0) n >>= 1; }
        }
        return (long)n << Math.Min(zm, zn);
    }
    /// <summary>
    /// 最大公約数
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    //public static BigInteger GCD(BigInteger a, BigInteger b) { return BigInteger.GreatestCommonDivisor(a, b); } //comment out if AOJ

    /// <summary>
    /// 最小公倍数
    /// </summary>
    /// <param name="n"></param>
    /// <param name="m"></param>
    /// <returns></returns>
    public static long LCM(long n, long m) { return Math.Abs((n / GCD(n, m)) * m); }

    /// <summary>
    /// n以下の素数のリストを返す (get all primes less than or equal to n)
    /// </summary>
    /// <param name="n"></param>
    /// <returns></returns>
    public static List<int> GetPrimes(int n)
    {
        if (n < 3) n = 3;
        var m = (n - 1) >> 1;
        var primes = new List<int>((int)(n / Math.Log(n)));
        primes.Add(2);
        var composites = new bool[m];
        composites[0] = false;
        for (var p = 0; p < m; p++)
        {
            if (!composites[p])
            {
                var pnum = 2 * p + 3;
                primes.Add(pnum);
                for (var k = 3 * p + 3; k < m; k += pnum) composites[k] = true;
            }
        }
        return primes;
    }

    #endregion

    #region Mod
    const int mod = 1000000007;

    public struct Mod
    {
        public long n;
        public Mod(long m)
        {
            n = m;
            if (n >= mod) n %= mod;
            else if (n < 0) n = (n % mod + mod) % mod;
        }

        public static explicit operator long(Mod m) { return m.n; }

        public static Mod operator +(Mod a, Mod b) { long tmp = a.n + b.n; if (tmp >= mod) tmp -= mod; return new Mod(tmp); }
        public static Mod operator -(Mod a, Mod b) { long tmp = a.n - b.n; if (tmp < mod) tmp += mod; return new Mod(tmp); }
        public static Mod operator *(Mod a, Mod b) { long tmp = ((a.n * b.n) % mod); return new Mod(tmp); }
        public static Mod operator ^(Mod a, int n)
        {
            if (n == 0) return new Mod(1);
            Mod res = (a * a) ^ (n / 2);
            if (n % 2 == 1) res = res * a;
            return res;
        }
        public static Mod operator /(Mod a, Mod b)
        {
            return a * new Mod(inv(b.n, mod));
        }
    }


    static long inv(long a, long p)
    {
        return (a == 1 ? 1 : (1 - p * inv(p % a, a)) / a + p);
    }

    const int MAX_N = 1024000;
    static bool modFact_Initialized = false;
    static Mod[] fact = new Mod[MAX_N], factinv = new Mod[MAX_N];
    static void init()
    {
        modFact_Initialized = true;
        fact[0] = new Mod(1); factinv[0] = new Mod(1);
        for (int i = 0; i < MAX_N - 1; i++)
        {
            fact[i + 1] = fact[i] * new Mod(i + 1);
            factinv[i + 1] = factinv[i] / new Mod(i + 1);
        }
    }
    /// <summary>
    /// aCb (combination, 組合せ)
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public static Mod comb(int a, int b)
    {
        if (modFact_Initialized == false) { init(); }
        return fact[a] * factinv[b] * factinv[a - b];
    }

    #endregion
}
#endregion

/// <summary>
/// 検証済みライブラリ
/// </summary>
static partial class Func
{
    /// <summary>
    /// 個数制限なしナップザック問題 O(NW)
    /// </summary>
    /// <param name="w">重さ</param>
    /// <param name="v">価値</param>
    /// <param name="W">ナップザックの容量</param>
    /// <returns></returns>
    public static int KnapsackWUnlimited(int[] w, int[] v, int W)
    {
        var N = w.Length;
        var dp = new int[W + 1];
        for (var i = 0; i < N; i++) for (var j = w[i]; j <= W; j++)
                dp[j] = Math.Max(dp[j], v[i] + dp[j - w[i]]);
        return dp[W];
    }
}