A number theory problem concerning effects of shifting digits

Question

Find a positive integer whose first digit is 1 and which has the property that, if this digit is transferred to the end of the number, the number is tripled.

A full proof will be much appreciated!

Answer 1

Two aproaches:

Often, and here, you can just construct it. You want $$3\cdot \overline {1abcd\ldots}=\overline{abcd\ldots 1}$$ You can see the last digit is $7$, so write it in on the right and find the next to last digit. Keep going until you find a $1$ on the left. Once you find one solution, you can repeat it as many times as you want.

Write your number as $10^n+b$ where $b$ has $n$ digits. Then the requirement is $$3\cdot(10^n+b)=10b+1\\7b=10^n-1$$ so find an $n$ so that $10^n-1$ is a multiple of $7$