Can't understand INOI problem "Periodic Strings"

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See title, here is the problem statement. Specifically I don’t understand what is meant my “Output Format - the number of non-periodic strings of length N , modulo M” bit. Could someone give me an example to explain the output format?

Here is the problem https://www.codechef.com/IOIPRAC/problems/INOI1502

I am also pasting this here in case someone doesn’t want to click the link.

A string is any nonempty sequence of 0s and 1s. Examples of strings are 00, 101, 111000,
1, 0, 01. The length of a string is the number of symbols in it. For example, the length of
111000 is 6. If u and v are strings, then uv is the string obtained by concatenating u and v.
For example if u = 110 and v = 0010 then uv = 1100010.
A string w is periodic if there exists a string v such that w = v
n = vv · · · v (n times),
for some n ≥ 2. Note that in this case the length of v is strictly less than that of w. For
example, 110110 is periodic, because it is vv for v = 110.
Given a positive integer N, find the number of strings of length N which are not periodic.
Report the answer modulo M. The non-periodic strings of length 2 are 10 and 01. The nonperiodic
strings of length 3 are 001, 010, 011, 100, 101, and 110.

Input format:
A single line, with two space-separated integers, N and M.

Output format:
A single integer, the number of non-periodic strings of length N, modulo M.

Test Data

In all subtasks, 2 ≤ M ≤ 108

The testdata is grouped into 4 subtasks.

Subtask 1 (10 marks) 1 ≤ N ≤ 4000. N is the product of two distinct prime numbers.

Subtask 2 (20 marks) 1 ≤ N ≤ 4000. N is a power of a prime number.

Subtask 3 (35 marks) 1 ≤ N ≤ 4000.

Subtask 4 (35 marks) 1 ≤ N ≤ 150000.

2

Non-Periodic means the strings which cannot be broken down into some recurring sub-string.

Lets take an example with N = 4

All possible periodic strings are:

0000, 0101, 1010, 1111

each with periodic sub-string ‘0’,‘01’,‘10’,‘1’ respectively.

Lets take an example of a non-periodic string 1110. You cannot have a sub-string of it which you can repeat 2 or more times to get original string.

So for N=4 your output should be 12%M. As there are 12 strings of length 4 which are non-periodic

2 Likes
3

it is impossible this problem. Gave a string from length N there are 2^N possible strings, which I have subtract the periodics one.
But I can’t compute 2^150000. Nobody can do it. There is something wrong. May be the problem requires the periodic one.

4

Rather than vote down, please explain your point of view. This is math.

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Not fully accomplishing. The problems require (2^N - k)%M and not 2^N%M -k . For example N=3 : 2^3 =8 - 2(periodic)=6 Than compute modulo for 6. How can get ride of it?