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 :
 * $A : B = D : E$