How long could it be? 2
Number Theory
Level
5
A positive integer ends with a 7 when written in decimal. If this 7 is moved to the front of the number, the resulting number is seven times of the original number. What is the smallest number that satisfies this property?
- This problem is based off How long could it be?
The answer is 1014492753623188405797.
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1 solution
1014492753623188405797 (22 digits)
When I saw this problem I first tried the number 142857, however this one increases not 7 times but rather 5 times. However I knew that this I needed to look for.
I then had an equation with n (our number without the last digit) and p (a power of 10 with the properties that p>n and p<10n. That equation was really that 7 times (10n+7) gave 7 p+n. That got equivalent to 69n+49=7 p. That number 69 was actually interesting, because it had to be a cycle in 1/69.
I calculated the decimal expansion of 1/69 (true, I used Wolfram Alpha). It gave me that 0144927536231884057971 * 69 = 9999999999999999999999. That wasn't exactly a hint.
For the sake of it, I tried 7 times that number, and noticed something: 1014492753623188405797 * 69 = 79999999999999999999993. The number got shifted exactly once as I multiplied by 7. If I multiplied this number with 7 again, the first digit would become 7 so I decided to try it. It worked!
This solution was only partially deterministic, but, well, I didn't need exact rigor to finish this!