Multiplication that preserves the digits

For 1. :

Yes. By the binomial theorem,

$$\begin{align}x \left( \frac{10^n}{x} + 1 \right)^m& =x\sum_{k=0}^{m}\binom mk\left(\frac{10^n}{x}\right)^k\\\\& =x\left(1+\sum_{k=1}^{m}\binom mk\left(\frac{10^n}{x}\right)^k\right)\\\\&=x+10^n\sum_{k=1}^{m}\binom mk\left(\frac{10^n}{x}\right)^{k-1} \\\\&\equiv x\pmod{10^n}\end{align}$$

Tags:

Number Theory

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