Problem 6

We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity. Find all positive integers

n

which have an alternating multiple.

Step 4 of 5: Classify

n

not divisible by

20

Previous step Next step

In plain words

Classify $n$ not divisible by $20$

n=2^u5^vm,\quad\gcd(m,10)=1,\quad20\nmid n\implies\begin{cases}n\mid2^{k+1}m& (v=0),\\ n\mid5^km&(u=0),\\ n\mid2\cdot5^km&(u=1,v\ge1),\end{cases}\quad k\ge2\text{ even}

Detailed analysis

Write $n=2^u5^vm$ with $\gcd(m,10)=1$ . Since $20\nmid n$ , either $v=0$ , or $u=0$ , or $u=1$ and $v\ge1$ . Choose an even $k\ge2$ large enough. In the three cases, respectively, $n$ divides $2^{k+1}m$ , $5^km$ , or $2\cdot5^km$ .