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?

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Breakdown:
Subject: Number Theory
Focus: Review Problem

1 Answer
Kha L.
Jul 31, 2018

#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

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