What number the letter O represents in this cryptarithm?

Question

I found a question and I do not know how to do it. The goes as: If letters all represent digits, what is the value of O in:

$$\begin{array}{ccccccc} &&&&O&N&E\\ +&&&F&O&U&R\\ \hline &&&F&I&V&E\\ \end{array}$$

and

$$\begin{array}{ccccccc} &&&&O&N&E\\ x&&&&&&4\\ \hline &&&F&O&U&R\\ \end{array}$$

What I know

I have figured out that the R is a zero and E is a 5. But then I do not know how to go on. I went with N being 3 in the second algorithm and found that I got stuck. I tried some more but that didn't work either.

Just Reference: The 4 can be repeated, no other number can be repeated unless it is the same number. Can I have some help, please?

Answer 1

1. From "ONE+FOUR=FIVE" one gets $R=0$, thus "ON+FOU=FIV", thus "ON+OU=IV", thus $1\leqslant O\leqslant4$.

2. From "4·ONE=FOU0" you get that $E\equiv 0 \pmod 5$.

3. Assuming different letters represent different figures, $E=5$ because 0 is already occupied by $R$.

4. 4·ON5=FOU0", thus $U$ is even, $F=1$ and

   • $O\neq1$ because otherwise "4·ON5" would by smaller than 1000.
   • $O \neq 4$ because otherwise $O\geqslant6$.
   • $O \neq 2$ because otherwise "12xy / 4 = 3wz" and hence $O=3$.

Hence $O=3$