Question

I am playing a walking game with myself. On move 1, I do nothing, but on move `n` where `2 \le n \le 25`, I take one step forward if `n` is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to cont. below?

my original starting point. How many steps long is my walk back?

~ Question from AoPS ~

Breakdown:
Subject: Number Theory
Focus: Review Problem

Answer 1

`21` steps

Explanation:

From integers `2` to `25`, there are `color(red)(9)` `color(red)(pri me)` numbers (`2, 3, 5, 7, 11, 13, 17, 19,` and `23`). That means the remaining `color(blue)(15` numbers are `color(blue)(composite)`. You take `color(green)(1)` step `color(green)(f o r w ard)` for every prime, and `color(purple)(2)` steps `color(purple)(backwards)` for every composite. We can write your position from your original point as:

`(color(green)1)(color(red)(9))+(color(purple)(-2))(color(blue)(15))=9-30=-21`

This is our displacement; we want `d i s t a n c e`, so we take the absolute value of `-21`.

`|-21|=21` steps