Prove that there exists no natural number such that shifting its first digit to the end, multiplies it by 5, 6, or 8.
If possible, let there exist such a natural number N having precisely D digits.
Suppose, the first digit of N exceeds 1.
Then, if we multiply N by 5,6, or 8- then the resultant product will have D+1 digits.
This is a contradiction.
Therefore, the first digit of N must be 1.
Now, shifting 1 from the first position to end will result in a number having the form M**....**1.
Obviously, a number having 1 as the units digit cannot be divisible by 5,6, or 8.
This is a contradiction.
Consequently, there does NOT exist any such natural number N in conformity with the given conditions.
Edited on May 11, 2022, 1:39 am
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