3. Aroon's PIN has four digits. When the first digit (reading from the left) is moved to the end of the PIN, the resulting integer is 6 less than 3 times Aroon's PIN. What could Aroon's PIN be?
SOlUTION
Let's denote Aroon's PIN by $1000 a+b$, where $0 \leq a \leq 9$ is its first digit and $0 \leq b<1000$ is the three-digit number formed by the remaining digits of his PIN.

When the first digit of Aroon's PIN is moved to the end, the new number is $10 b+a$. Hence:
$$
\begin{array}{c}
3(1000 a+b)-6=10 b+a \\
\therefore 2999 a=7 b+6
\end{array}
$$

Given $0 \leq b<1000$, we have $6 \leq R H S<7006$, hence $0<a<3$, so $a$ must be 1 or 2 .
If $a=1$, we have $7 b+6=2999$, so $b=427 \frac{4}{7}$, which is not a whole number.
If $a=2$, we have $7 b+6=5998$, so $b=856$. We can easily check that $2856 \times 3-6=8562$. Hence Aroon's PIN must be 2856 .