Doubling an integer by moving one digit?

Question

x is a positive integer. If the rightmost digit of x is moved to the leftmost position (for example 1234 would become 4123), the new integer is exactly 2x. As you might expect, there are an infinite number of integers with this property. What is the smallest one?

Answer 1

Consider zero, zero.
00 = 0
0*2 = 0
If you except double zero to be equal to zero, then this could be your answer.

Since the numbers are being switched around, then you need at least two numbers.
Let p and q be these unknown numbers.
So x = [p,q] and 2x = [q,p].
2[p,q] = [q,p] But p is a tens unit, and q is a units unit.
2[10p,q] = [q,10p]
[20p,2q] = [q,10p] Match up the tens and units together.

20p = q eq.1
2q = 10p Substitute for q.
---------------
2p = 10p
2(20p) = 10p
40p = 10p ( ÷ 10 )
4p = p
This implies, p = 0.

20p = q eq.1
20(0) = q
→ q = 0.

The lowest number could be taken as 00.

Answer 2

No such integer exists; I can prove this but think it is waste to show the proof here. If you are seriously interested in the proof mail me.

Answer 3

I know what a digit is, but whats an integer? DUH?