Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer [closed]
Solution 1:
$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.$$ If this is an integer then in particular $b \mid ad \implies b\mid d$, and viceversa $d\mid b$. This condition implies that $b=\pm d$. But they are both positive, therefore they have to be equal. Contradiction.
Solution 2:
Hint $ $ If $\,\ \dfrac{a}b + \dfrac{c}{d}\, =\, n\, $ then $\ \dfrac{a}{b}\, =\, \dfrac{dn-c}d.\ $ Both are in lowest terms $\,(p\mid d,\,dn\!-\!c\,\Rightarrow\,p\mid c)\,$ therefore $\ b = d\ $ by the $ $ uniqueness of reduced fractions ("unique fractionization").