With a calculator, I have noticed that the integer $37$ multiplied with some particular numbers yields numbers with some structures. 

For instance, let $aaaa\ldots a$ be a natural number of $n$ identical digits. Then, $ 37 \cdot aaaa\ldots a$ is a number with $n+1$ or $n+2$ digits of the form

> 
> $$\underbrace{4\cdot a}\_{1\text{ or }2}~ \underbrace{aaa\ldots a}\_{n-2} ~\underbrace{7\cdot a}\_{2^{\*}}.$$
> 
> 
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$\*$ if $a=1$, then $7 \cdot a$ is the sequence of digits $07$. 

I am wondering whether the result above can be proven using some number-theory tool.

Thanks in advance!