## Proposition 36

 To find the least number which three given numbers measure. Let A, B, and C be the three given numbers. It is required to find the least number which they measure. Take D the least number measured by the two numbers A and B. VII.34 Then C either measures, or does not measure, D. First, let it measure it. But A and B also measure D, therefore A, B, and C measure D. I say next that it is also the least that they measure. If not, A, B, and C measure some number E less than D. Since A, B, and C measure E, therefore A and B measure E. Therefore the least number measured by A and B also measures E. VII.35 But D is the least number measured by A and B, therefore D measures E, the greater the less, which is impossible. Therefore A, B, and C do not measure any number less than D. Therefore D is the least that A, B, and C measure. Next, let C not measure D. Take E, the least number measured by C and D. VII.34 Since A and B measure D, and D measures E, therefore A and B also measure E. But C also measures E, therefore A, B, and C also measure E. I say next that it is also the least that they measure. If not, A, B, and C measure some number F less than E. Since A, B, and C measure F, therefore A and B measure F. Therefore the least number measured by A and B also measures F. But D is the least number measured by A and B, therefore D measures F. But C also measures F, therefore D and C measure F, so that the least number measured by D and C also measures F. VII.35 But E is the least number measured by C and D, therefore E measures F, the greater the less, which is impossible. Therefore A, B, and C do not measure any number which is less than E. Therefore E is the least that is measured by A, B, and C. Q.E.D.
The least common multiple of three numbers a, b, and c can be found as

LCM(a, b, c) = LCM(LCM(a, b), c)).

This proposition is used in the proof of proposition VII.39.

Next proposition: VII.37

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 Select from Book VII Book VII intro VII.Def.1-2 VII.Def.3-5 VII.Def.6-10 VII.Def.11-14 VII.Def.15-19 VII.Def.20 VII.Def.21 VII.Def.22 VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13 VII.14 VII.15 VII.16 VII.17 VII.18 VII.19 VII.20 VII.21 VII.22 VII.23 VII.24 VII.25 VII.26 VII.27 VII.28 VII.29 VII.30 VII.31 VII.32 VII.33 VII.34 VII.35 VII.36 VII.37 VII.38 VII.39 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip