Hi , let's think about this problem :
Find the smallest natural number with as the last digit , such that if the final is moved to the front of the number it is multiplied by .
Is the answer?Spoiler:
Hi , let's think about this problem :
Find the smallest natural number with as the last digit , such that if the final is moved to the front of the number it is multiplied by .
Is the answer?Spoiler:
Let the number be .
Let .
Then .
Therefore .
By backwards argument it's easy to see that all the cycles in are solutions.
Thus we have found all the solutions, the smallest of which is .
This technique can be applied to this kind of problems in many cases.
Hello, simplependulum!
Just write out the multiplcation,
. . and the digits will appear sequentially.
Find the smallest natural number with as the last digit,
such that if the final is moved to the front of the number it is multiplied by .
We have: .
In column-6: .
. .
In column-5: .
. .
In column-4: .
. .
In column-3: .
. .
In column-2: .
Therefore: .
Fyi, there is a name for these, n-parasitic number, where here n=4 and k=6.