Find the value of a if A divisable by 3

Question

$A=0.\overline{a}+0.\overline{aa}+0.\overline{aaa}\: \\ \text{If A is divisable by 3, }\: \text{find the value of} \: a\: $

Answer 1

We know that$$A=a(0.1+0.11+0.111)$$notice that since $3|0.1+0.11+0.111$ we have $3|A$ therefore $a$ can be anything

Answer 2

Assuming $a$ is single digit natural number (including zero).

$$ A=0.\overline{a}+0.\overline{aa}+0.\overline{aaa}\: $$

Multiplying both sides by 10: $$ 10A=a.\overline{a}+a.\overline{aa}+a.\overline{aaa}\: $$

Subtracting both equations we get: $$ 9A=a+a+a =3a $$ $$ 3A=a $$ Thus either $(A=0)\wedge(a=0)$ or $(A=3)\wedge(a=9)$.

Answer 3

Assuming you mean decimal representation, and assuming $a\in\lbrace0,1,2,3,4,5,6,7,8,9\rbrace$.

We have : $$0.\overline{a}=a/10$$ $$0.\overline{aa}=a/10+a/100=11a/100$$ $$0.\overline{aaa}=a/10+a/100+a/1000=111a/1000$$

Hence : $$A=(111+110+100)a/1000=321a/1000$$

The thing is : $A$ has to be an integer to be able to say that "$3$ divides $A$", which is clearly not the case...

Hope that helps.