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By Jan C. A. Boeyens, Demetrius C. Levendis

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Each d|a, therefore a = d pa11−d1 pa22 −d2 . . par r −dr No other integers divide a, since4 pgi i ∤ ai if gi > ai or if pi is not one of the primes in the factorization of a. From this observation, pc11 pc22 . . pcrr divides both a and b and is in turn divided by every common divisor. 4 ∤, does not divide 32 CHAPTER 2. NUMBER THEORY PRIMER Corollary Two positive integers are relatively prime iff5 their factorizations have no primes in common. [a, b], the least common multiple (LCM) of a and b is a multiple of a and a multiple of b and it divides every common multiple of a and b.

7) For a given prime its multiples in common with those of lower primes occur as cm with 6m < p. 2 how this procedure screens out composite numbers less than 300. The procedure outlined here may be used as an algorithm to identify prime numbers, but not as a formula to generate them. 9: The prime numbers ≥5 and their common multiples arranged on a 14-period spiral. 3. 2: Composite numbers of the form p(6m ± 1) < 300. p m 5 1 2 3 4 − 25 55 85 115 145 175 205 235 265 295 + 35 65 95 125 155 185 215 245 275 − 77 119 161 203 245 287 + 49 91 133 175 217 259 − 121 187 253 + 143 209 275 − 221 299 7 11 13 + 5 6 7 8 9 10 169 247 − 17 + 289 only multiples of p, but also the common multiples of all other primes.

From this observation, pc11 pc22 . . pcrr divides both a and b and is in turn divided by every common divisor. 4 ∤, does not divide 32 CHAPTER 2. NUMBER THEORY PRIMER Corollary Two positive integers are relatively prime iff5 their factorizations have no primes in common. [a, b], the least common multiple (LCM) of a and b is a multiple of a and a multiple of b and it divides every common multiple of a and b. Using arguments like those in the derivation of (a, b) it is shown that mr 1 m2 [a, b] = pm 1 p2 .

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