Proposition VII.35 — If two numbers measure any number, the least number measured…

Proof

Let

L = lcm (a, b)

and suppose

a, b ∣ n

. Divide

n

L

with remainder; the remainder

r

is measured by both

a

and

b

(Common Notion 3 on the equimultiples) and is less than

L

. By minimality of

L

r = 0

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Full neighborhood

Depends on (2)

VII.34Proposition VII.34Given two numbers, to find the least number which they measure.
3Common notion 3If equals be subtracted from equals, the remainders are equal.

Required by (dependents) (1)

VII.36Proposition VII.36Given three numbers, to find the least number which they measure.

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