Book:Euclid/The Elements/Book VII

Euclid: The Elements: Book VII

Published $\text {c. 300 B.C.E}$

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 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

• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $7$: Patterns in Numbers: Euclid