Book:Euclid/The Elements/Book VII

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Euclid: The Elements: Book VII

Published $c. 300 B.C.E$.


Contents

Book $\text{VII}$: Number Theory

Definitions
Proposition $1$: Sufficient Condition for Coprimality
Proposition $2$: Greatest Common Divisor of Two Numbers (Euclidean Algorithm)
Porism to Proposition $2$: Greatest Common Divisor of Two Numbers (Euclidean Algorithm)
Proposition $3$: Greatest Common Divisor of Three Numbers
Proposition $4$: Natural Number Divisor or Multiple of Divisor of Another
Proposition $5$: Divisors obey Distributive Law
Proposition $6$: Multiples of Divisors obey Distributive Law
Proposition $7$: Subtraction of Divisors obeys Distributive Law
Proposition $8$: Subtraction of Multiples of Divisors obeys Distributive Law
Proposition $9$: Alternate Ratios of Equal Fractions
Proposition $10$: Multiples of Alternate Ratios of Equal Fractions
Proposition $11$: Proportional Numbers have Proportional Differences
Proposition $12$: Ratios of Numbers is Distributive over Addition
Proposition $13$: Proportional Numbers are Proportional Alternately
Proposition $14$: Proportion of Numbers is Transitive
Proposition $15$: Alternate Ratios of Multiples
Proposition $16$: Natural Number Multiplication is Commutative
Proposition $17$: Multiples of Ratios of Numbers
Proposition $18$: Ratios of Multiples of Numbers
Proposition $19$: Relation of Ratios to Products‎
Proposition $20$: Ratios of Fractions in Lowest Terms
Proposition $21$: Coprime Numbers form Fraction in Lowest Terms
Proposition $22$: Numbers forming Fraction in Lowest Terms are Coprime
Proposition $23$: Divisor of One of Coprime Numbers is Coprime to Other
Proposition $24$: Integer Coprime to all Factors is Coprime to Whole
Proposition $25$: Square of Coprime Number is Coprime
Proposition $26$: Product of Coprime Pairs is Coprime
Proposition $27$: Powers of Coprime Numbers are Coprime
Proposition $28$: Numbers are Coprime iff Sum is Coprime to Both
Proposition $29$: Prime not Divisor implies Coprime
Proposition $30$: Euclid's Lemma for Prime Divisors
Proposition $31$: Composite Number has Prime Factor
Proposition $32$: Natural Number is Prime or has Prime Factor
Proposition $33$: Least Ratio of Numbers
Proposition $34$: Existence of Lowest Common Multiple
Proposition $35$: LCM Divides Common Multiple
Proposition $36$: LCM of Three Numbers
Proposition $37$: Integer Divided by Divisor is Integer
Proposition $38$: Divisor is Reciprocal of Divisor of Integer
Proposition $39$: Least Number with Three Given Fractions


Sources