Question

Replace $*$ in $312*7$ by the smallest number so that the number formed is divisible by $3$ and not by $9$.

Answer 1

Divisiblity rule for $3$ = if the sum of digits of a number is divisible by $3$

Divisibility rule for $9$ = if the sum of digits of a number is divisible by $9$

The number will be divisible by $3$ only when sum of digits is also divisible by $3$ but the sum of digits must not be divisible by $9$ as per question.

So, sum of digits of a number = $3+1+2+*+7=13+*$

Next number greater than $13$ which is a multiple of $3=15$

$15$ is also not divisible by $9$

hence required number =

$$\begin{gathered} =13+*=15 \\ *=15-13 \\ *=2 \end{gathered}$$