Proposition VII.1 — Two unequal numbers being set out, and the less being contin…

Proof

Suppose for contradiction some number

d > 1

measures both inputs. At each subtraction step the divisor and remainder differ from the previous pair by a common multiple of

d

; so

d

persists through the algorithm and ultimately measures the unit, which is impossible.

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Depends on (2)

VII.1Definition VII.1A unit is that by virtue of which each of the things that exist is called one.
VII.11Definition VII.11A prime number is that which is measured by a unit alone.

Required by (dependents) (1)

VII.2Proposition VII.2Given two numbers not prime to one another, to find their greatest common measure.

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