Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers will be prime to one another.
For, the less of two unequal numbers AB, CD being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left;
I say that AB, CD are prime to one another, that is, that an unit alone measures AB, CD.
For, if AB, CD are not prime to one another, some number will measure them.
Let a number measure them, and let it be E;
let CD, measuring BF, leave FA less than itself,
let AF, measuring DG, leave GC less than itself,
and let GC, measuring FH, leave an unit HA.
Since, then, E measures CD, and CD measures BF, therefore E also measures BF.
But it also measures the whole BA; therefore it will also measure the remainder AF.
But AF measures DG; therefore E also measures DG.
But it also measures the whole DC therefore it will also measure the remainder CG.
But CG measures FH; therefore E also measures FH.
But it also measures the whole FA; therefore it will also measure the remainder, the unit AH, though it is a number: which is impossible.
Therefore no number will measure the numbers AB, CD; therefore AB, CD are prime to one another. [VII. Def. 12]
Q. E. D.
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Book I