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Problem 6

Let p,q,n p,q,n be positive integers with p+q<n p+q<n . Let x0,x1,…,xn x_0,x_1,\ldots,x_n be integers with x0=xn=0 x_0=x_n=0, and for each 1≤i≤n1\le i\le n let xi−xi−1=p x_i-x_{i-1}=p or −q-q . Show that there exist i<j i<j , with (i,j)≠(0,n)(i,j)\ne(0,n), such that xi=xj x_i=x_j .
Step 2 of 4: Find the block length
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In plain words

Find the block length

n=k(p+q),k>1n=k(p+q),\quad k>1
Detailed analysis

Coprimality gives p∣s p\mid s ; write s=kp s=kp and therefore r=kq r=kq . Thus n=k(p+q) n=k(p+q). Since p+q<n p+q<n , k>1 k>1. Put h=p+q h=p+q .