Ratios of Multiples of Numbers

Theorem

 * If two (natural) numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.

Proof
Let two (natural) numbers $A, B$ by multiplying any number $C$ make $D, E$.

Then we need to show that $A : B = D : E$.


 * Euclid-VII-18.png

We have that $A \times C = D$.

So from Natural Number Multiplication is Commutative, also $C \times A = D$.

For the same reason, $C \times B = E$.

Therefore from, $A : B = D : E$.