Number ending in 4 quadrupled by rotating that digit

Find the smallest positive integer whose last digit is $4$ and which becomes exactly four times larger when that last digit is moved to the front (for example, $1234$ would become $4123$).

$102564$. Build it digit by digit from the right: multiply by $4$ and carry, feeding each product digit into the next place, until a $4$ appears with no carry. This gives $102564$, and indeed $4\times102564=410256$.