# Book:Euclid/The Elements/Book VII

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## Euclid:

## Euclid: *The Elements: Book VII*

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

### Contents

Book $\text{VII}$: Number Theory

- Proposition $1$: Sufficient Condition for Coprimality
- 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

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