Prove there is no natural number $n$ such that $n$ is $2,5,6$ or $8$ times by the power of its units digit.

Question

One of my friend showed me yesterday this 5th grade problem.

For me the incoherence and the language is a bit strange, but I do believe that with your help we can manage to demonstrate this:

Show that there isn't a natural number which increased $2,5,6$ or $8$ times by the power of its first digit at the end of the number.

Answer 1

We can consider for $x_i=\{0,1,...,9\}$

$$N=10^n x_n+...+10x_1+x_0$$

$$N^{x_0}-N=kN\implies N^{x_0-1}-1=k\implies N^{x_0-1}=k+1$$

then

• $x_0=1\implies k=0$
• $x_0=2\implies k=1,11,...$
• $x_0=3\implies k=8,168,...$
• $x_0=4\implies k=64,2743,...$
• $x_0=5\implies k=624,...$
• $x_0=6\implies k=7775,...$
• ...