Coursework.Info - Inspiration in an Instant

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Introduction The Phi is described as the number of positive integers less than n (positive integer) which have no factor, other than 1, in common, co-prime, with n. In my investigation I aim to identify and explain the rules and equations associated with the Phi function. To go about this I will investigate ? (n). This will be further touched upon and will help me investigate and support any conclusions I hope to gain from investigating whether (a x b) = (a) x (b) in certain cases. Table of Phi's from 2-40 Number Phi ?2 1, 1 ?3 1,2 2 ?4 1,2,3 2 ?5 1,2,3,4 4 ?6 1,2,3,4,5 2 ?7 1,2,3,4,5,6 6 ?8 1,2,3,4,5,6,7 4 ?9 1,2,3,4,5,6,7,8 6 ?10 1,2,3,4,5,6,7,8,9 4 ?11 1,2,3,4,5,6,7,8,9,10 10 ?12 1,2,3,4,5,6,7,8,9,10,11 4 ?13 1,2,3,4,5,6,7,8,9,10,11,12 12 ?14 1,2,3,4,5,6,7,8,9,10,11,12,13 6 ?15 1,2,3,4,5,6,7,8,9,10,11,12,13,14 8 ?16 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 8 ?17 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 16 ?18 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 6 ?19 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18 18 ?20 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19 8 ?21 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 12 ?22 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21 10 ?23 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21,22 22 ?24 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21,22,23 8 ?25 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24 20 ?26 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 12 ?27 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 18 ?28 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 12 ?29 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28 28 ?30 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29 8 ?31 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30 30 ?32 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31 16 ?33 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32 20 ?34 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33 16 ?35 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34 24 ?36 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35 12 ?37 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36 36 ?38 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37 18 ?39 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38 24 ?40 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, 21, 22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39 16 ?41 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40 40 ?42 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41 12 ?43 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42 42 ?44 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43 20 ?45 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44 24 ?46 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45 22 ?47 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46 46 ?48 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47 16 ?49 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47,48 42 ?50 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47,48,49 20 ?51 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50 32 ?52 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51 24 ?53 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52 52 ?54 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52,53 18 ?55 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, 23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40, 41,42,43,44,45,46,47,48,49,50,51,52,53,54 40 In my coursework there are...

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