PROPOSITION 7.

If a composite number by multiplying any number make some number, the product will be solid.

For let the composite number A by multiplying any number B make C;

I say that C is solid.

A B C D E
For, since A is composite,
it will be measured by some number. [VII. Def. 13]

Let it be measured by D; and, as many times as D measures A, so many units let there be in E.

Since then D measures A according to the units in E,
[VII. Def. 15]therefore E by multiplying D has made A.
And, since A by multiplying B has made C, and A is the product of D, E,
therefore the product of D, E by multiplying B has made C.
Therefore C is solid, and D, E, B are its sides.
Q. E. D.

Dependency Graph

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See Commentary on Proposition IX.7.