# 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.

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.