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Question Number 80974 by Rio Michael last updated on 08/Feb/20

 Show that gcd (a , a + x) ∣ x  hence show that any two consecutive  integers are coprime

$$\:\mathrm{Show}\:\mathrm{that}\:\mathrm{gcd}\:\left({a}\:,\:{a}\:+\:{x}\right)\:\mid\:{x} \\ $$$${hence}\:{show}\:{that}\:{any}\:{two}\:{consecutive} \\ $$$${integers}\:{are}\:{coprime} \\ $$

Commented by kaivan.ahmadi last updated on 09/Feb/20

if gcd(a,a+x)=d⇒d∣a⇒a=dy  and d∣a+x⇒a+x=dt  ⇒dy+x=dt⇒x=dt−dy=d(t−y)⇒d∣x

$${if}\:{gcd}\left({a},{a}+{x}\right)={d}\Rightarrow{d}\mid{a}\Rightarrow{a}={dy} \\ $$$${and}\:{d}\mid{a}+{x}\Rightarrow{a}+{x}={dt} \\ $$$$\Rightarrow{dy}+{x}={dt}\Rightarrow{x}={dt}−{dy}={d}\left({t}−{y}\right)\Rightarrow{d}\mid{x} \\ $$

Commented by Rio Michael last updated on 09/Feb/20

thanks

$${thanks} \\ $$

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