Math Gold Medalist

Lor

2023 AIME II 

Problem 15

For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n \equiv 1 \pmod{2^n}.$ Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n = a_{n+1}.$

Small Example

Modular Arithmetic

 

Calculate a1, a2, a3, a4

a1=23

a2=69

a3=161

a4=161

Then generalize

Calculate powers of 2 mod 23

Consider n mod 11

.

   Solution