---
book: 7
number: 35
id: "VII.35"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.35"
license: "CC-BY-SA-4.0"
---

# VII.35

*If two numbers measure any number, the least number measured by them will also measure the same.*

## Proof

For let the two numbers *A*, *B* measure any number *CD*, and let *E* be the least that they measure; I say that *E* also measures *CD*.

For, if *E* does not measure *CD*, let *E*, measuring *DF*, leave *CF* less than itself.

Now, since *A*, *B* measure *E*, and *E* measures *DF*, therefore *A*, *B* will also measure *DF*.

But they also measure the whole *CD*; therefore they will also measure the remainder *CF* which is less than *E*: which is impossible.

Therefore *E* cannot fail to measure *CD*; therefore it measures it. Q. E. D.