TITLE: What's the smallest number with first digit 1 that triples when this digit is moved to the end?
QUESTION [32 upvotes]: There's this homework question I have, and while I know people generally don't like these, I would like a hint on how to get started please.
A positive integer begins with the digit 1 when written in decimal. When this digit is transferred to the end of the number, the number is tripled. Find the smallest number that has this property.
I know it has something to do with divisibility by 3.
How should I start? Could someone please give an insightful comment that might let me solve it (without feeling like cheating)?
Edit: I suppose that transfer means that 12345 would become 23451...

REPLY [6 votes]: Here's a slightly different solution:
We must have
$$
10x + 1 = 3(x+10^k)
$$
for some non-negative integers $x$ and $k$.
Simplifying, we get
$$
7x = 3\times 10^k - 1
$$
so
$$
3\times 10^k \equiv 1 (\mod 7)
$$ 
or 
$$
3 ^ {k+1} \equiv 1 (\mod 7)
$$
Working mod 7, we have $3^2\equiv2$, $3^3\equiv6$, $3^4\equiv4$, $3^5\equiv5$ and $3^6\equiv1$, so the minimal $k$ is 5. 
Thus 
$$
7x = 3\times10^5-1 = 299,999
$$
and $x=42,857$; the numbers are $428,571 = 3 \times 142,857$.