I don't know if this is the correct forum but...

Suppose you had a difference of fractions $\displaystyle 1/4 - 1/6$

You could get the LCD by factoring each denominator. For 4 you would have (2,2) for 6 you would have (2,3). This gives 12 as the LCD.

You would then multiply the numerator of $\displaystyle 1/4$ by 3, which is the factorization of 6 (2,3) not including the factor that is a factor of 4, namely 2. This gives $\displaystyle 3/12$

This prime number (3) is the same as dividing the LCD by the original denominator: $\displaystyle 12/4 = 3$

I hope this makes sense.

Is there a mathematical principle or law that describes this?

Thanks!