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G.255 number part parts — Heath¶

Any number is either a part or parts of any number, the less of the greater.

Let A, / B C / be two numbers, and let / B C / be the less ;

I say that / B C / is either a part, or parts, of A.

For A, / B C / are either prime to one another or not.

First, let A, / B C / be prime to one another.

Then, if / B C / be divided into the units in it, each unit of those in / B C / will be some part of A ;

so that / B C / is parts of A.

Next let A, / B C / not be prime to one another ; then / B C / either measures, or does not measure, A.

If now / B C / measures A, / B C / is a part of A.

But, if not, let the greatest common measure D of A, / B C / be taken ; [G.251]

and let / B C / be divided into the numbers equal to D, namely / B E /, / E F /, / F C /.

Now, since D measures A, D is a part of A.

But D is equal to each of the numbers / B E /, / E F /, / F C / ;

therefore each of the numbers / B E /, / E F /, / F C / is also a part of A ;

so that / B C / is parts of A.

Therefore etc. Q. E. D.

Dependency Graph¶