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Question Number 124693 by mathocean1 last updated on 05/Dec/20

$$Demonstratethat\mathrm{\forall }a,b\in {\mathbb{N}}^{\ast }if\frac{a}{b}$$$$cannotbesimplified,then\frac{a+b}{a^2+ab+b^2}$$$$cannotalsobesimplified.$$$$$$

Answered by MJS_new last updated on 05/Dec/20

$$\frac{a}{b}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{simplified}\iff a\ne bp\wedge b\ne aq$$$$\frac{a+b}{a^2+ab+b^2}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{simplified}\iff$$$$\iff \frac{a^2+ab+b^2}{a+b}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{simplified}$$$$\frac{a^2+ab+b^2}{a+b}=a+\frac{b^2}{a+b}=b+\frac{a^2}{a+b}$$$$\frac{b^2}{a+b}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{simplified}\mathrm{if}a\ne bp$$$$\frac{a^2}{a+b}\mathrm{can}\mathrm{not}\mathrm{be}\mathrm{simplified}\mathrm{if}b\ne aq$$

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