Example 1 Find the smallest natural number with the following properties:
(1) Its decimal representation ends with the digit 6;
(2) If the last digit 6 is deleted and this digit 6 is written in front of the remaining digits, then the resulting number is 4 times the original number.
(4th IM)
Solution: From (1), we know that the last digit of the required smallest natural number is 6. Combining (2) with the multiplication table, it is not difficult to know that its tens digit is 4, and it is also the last digit of the obtained number.
$46 \times 4=184$.
Thus, the tens digit of the obtained number should be 8, and it is also the hundreds digit of the required number.
$846 \times 4=3384$.
Thus, the hundreds digit of the obtained number should be 3, and it is also the thousands digit of the required number.
$3846 \times 4=15384$.
Thus, the thousands digit of the obtained number should be 5, and it is also the ten-thousands digit of the required number.
$53846 \times 4=215384$.
Thus, the ten-thousands digit of the obtained number should be 1, and it is also the hundred-thousands digit of the required number. Therefore,
$153846 \times 4=615384$.
In this way, the last digit 6 of the required number has appeared for the first time at the beginning of the obtained number, which meets the requirements. Such natural numbers also include $153846153846, \cdots$. Therefore, the smallest required natural number is 153846.