UKMT Putting A 1 At Beginning / End Of Number

I have a number that is less than one million. Putting a 1 after it
makes it three times as large as putting a 1 before it. What is my
number?

My workings:

My thought is that if something multiplied by 3 ends in a 1, it must start with a 7.
So I put a 7 in the result as well (because they're the same number) but now if I carry the 2 from 7 * 3 = 21 then I get 3*x + 2 = 7 and this has no integer solutions for x, what do I do now?

My thought is that if something multiplied by 3 ends in a 1, it must start with a 7.
So I put a 7 in the result as well (because they're the same number) but now if I carry the 2 from 7 * 3 = 21 then I get 3*x + 2 = 7 and this has no integer solutions for x, what do I do now?

3x + 2 = 7 has no integer solutions but the number doesn't need to be 7, just end in a 7. So 3x + 2 = (some number ending in 7). Does that help?

Putting your problem into maths, I thought of the answer being the solution to $3(x+10^n)=10x+1$ where $n \in \mathbb{Z^+}|n \leq 6$.

So $\displaystyle x=\frac{3\times10^n-1}{7}$. The thing on the RHS is $2\underbrace{99...99}_\text{n times}$. You keep doing long division into $7$ until you reach $n=5$ and you stop as you get an integer (you could check $n=6$ doesn't give an integer if you like). And yeah you get the same answer.

I'll be annoying and say "by inspection, 42857".

Slightly less annoying (and something that although rarely useful, is occasionally, worth knowing):

$1/7 = 0.\dot{1}4285\dot{7}$

If you write 142857 twice, we can read off the first 6 multiples by reading 6 sequential digits from the appropriate place:

$1^14^32^28^65^47^5142857$ (superscripts are to show where to start, not to indicate powers or anything):

E.g. to find 3 x 142857, start from the number with a superscript 3 to get: 428571

Also, 7 * 142857 = 999999

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