When is the lcm of a fraction sum the actual denominator.Consider a sumab+cd=xywhere each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we haveab+cd=a⋅d(b,d)+c⋅b(b,d)bdwhere (b,d) denotes the gcd and [b,d] denotes the lcm. My question is, when is it true that y=[b,d]? For example, this does not hold for56+114=3842=1921where 21≠ [14,6]Are there any simple necessary and sufficient conditions for y=[b,d]?Edit It has been suggested thaty=[b,d]⇔(b,d)=1is a necessary and sufficient condition. I'm interested in either a proof or a counterexample if possible.Edit 2 After a bit of searching, I've found124+116=548as a counterexample.I am still looking for nice conditions for this to be satisfied and I feel that I should give an explanation of exactly what type of condition I am seeking. Angela has provided a necessary and sufficient condition, but it does not seem to be "simpler" than simply adding the fraction and seeing if it reduces. This is perhaps ambitious, but I am looking for a condition which is simple enough to use by inspection for simple fractions.

Question

When is the lcm of a fraction sum the actual denominator.Consider a sumab+cd=xywhere each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we haveab+cd=a⋅d(b,d)+c⋅b(b,d)bdwhere (b,d) denotes the gcd and [b,d] denotes the lcm. My question is, when is it true that y=[b,d]? For example, this does not hold for56+114=3842=1921where 21≠ [14,6]Are there any simple necessary and sufficient conditions for y=[b,d]?Edit It has been suggested thaty=[b,d]⇔(b,d)=1is a necessary and sufficient condition. I&#039;m interested in either a proof or a counterexample if possible.Edit 2 After a bit of searching, I&#039;ve found124+116=548as a counterexample.I am still looking for nice conditions for this to be satisfied and I feel that I should give an explanation of exactly what type of condition I am seeking. Angela has provided a necessary and sufficient condition, but it does not seem to be &quot;simpler&quot; than simply adding the fraction and seeing if it reduces. This is perhaps ambitious, but I am looking for a condition which is simple enough to use by inspection for simple fractions.

Jamiya Bradford · Accepted Answer

let x=gcd(b,d)b=xed=xfab+cd=af+ceefxefx=lcm(b,d)e,f are relative &#039;sThe fraction simplifies when Undefined control sequence cancelUndefined control sequence cancel  which is when Undefined control sequence cancelUndefined control sequence cancel

When is the lcm of a fraction sum

Answered question

Answer & Explanation

New Questions in Pre-Algebra