Proposition 35

If two numbers measure any number, then the least number measured by them also measures the same.

Let the two numbers A and B measure any number CD, and let E be the least that they measure.

I say that E also measures CD.

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If E does not measure CD, let E, measuring DF, leave CF less than itself.

Now, since A and B measure E, and E measures DF, therefore A and B also measure DF. But they also measure the whole CD, therefore they measure the remainder CF which is less than E, which is impossible.

Therefore E cannot fail to measure CD. Therefore it measures it.

Therefore, if two numbers measure any number, then the least number measured by them also measures the same.

Q.E.D.

Guide

Outline of the proof

Assume both a and b divide c. Let e be their least common multiple. Suppose that e does not divide c. Then repeatedly subtract e from c to get c = ne + f, where the remainder f is less than e. Since a and b both divide c and e, they also divide f making f a smaller common multiple than the least common multiple e, a contradiction. Thus the least common multiple also divides c.

Use of this proposition

This proposition is used in the next one and in VIII.4.