## Sequence and divisibility

For discussing Olympiad Level Algebra (and Inequality) problems

### Sequence and divisibility

Let $n$ be a positive integer and let $a_1,a_2,a_3,\ldots,a_k$ $( k\ge 2)$ be distinct integers in the set ${ 1,2,\ldots,n}$ such that $n$ divides $a_i(a_{i + 1} - 1)$ for $i = 1,2,\ldots,k - 1$. Prove that $n$ does not divide $a_k(a_1 - 1).$
Katy729

Posts: 33
Joined: Sat May 06, 2017 2:30 am

### Re: Sequence and divisibility

For the sake of the contradiction, let's assume that it does. Then,
$a_1 \equiv a_1. a_2 \equiv a_1. a_2. a_3$ $\equiv........\equiv a_1. a_2......a_{k-2}. a_k$
$\equiv.....\equiv a_1. a_k \equiv a_k \pmod n$.

So, $n$ divides $|a_1 - a_k|$. But $0 < |a_1 - a_k| < n$, which is a contradiction.
So, $n$ doesn't divide $a_k(a_1 - 1)$.
I like girls and mathematics; both are beautiful.
tanmoy

Posts: 278
Joined: Fri Oct 18, 2013 11:56 pm

### Re: Sequence and divisibility

Thanks tanmoy.
Katy729

Posts: 33
Joined: Sat May 06, 2017 2:30 am