Help solving a digit and word code problem/riddle: sum of four "ONE"s is "TEN" 
Replace each letter by a digit. The same digit must represent each
letter, and no beginning letter of a word can be zero. No two letters can
be the same number. Find the digits represented by the letters 'O',
'N', 'E', 'T'.
O N E
O N E
O N E
O N E
=====
T E N

I have tried this, but I can't seem to crack it. It doesn't make sense to me. Can anyone help with this?
 A: A slightly different method:  EN has to be a multiple of 4 and EN+NE is a multiple of 5 (last two digits match 5×NE), therefore E+N is a multiple of 5.  For any choice of E we can pick N forcing the sum to be a multiple of 5 and N to be even, leaving only one possible N for each E.  The only such candidates for EN that are also multiples of 4:  00, 28, 32, 64, 96.  Of these only EN=00 and EN=28 are consistent with NE×4 ending with EN.  Then 00 violates the usual rule that different letters represent different digits and is just plain inelegant (100+...+100=400?  Come on!).  So E=2 and N=8.  We then can't begin ONE with 0 (convention) or greater than 1 (ONE<1000/4=250), so 182+...+182=728 is all there is.
A: If we assume $E$ and $N$ represent different digits, we need $E\ne0$.
$$400O+40N+4E=100T+10E+N$$
Therefore
$$
400O+39N=100T+6E
$$
If we reduce modulo $100$, we see that $39N\equiv 6E\pmod{100}$, so $13N\equiv 2E\pmod{100}$. Since the inverse of $13$ modulo $100$ is $77$, we have $N\equiv77\cdot2E\equiv54E\pmod{100}$.
Compute $54x$ modulo $100$ for $1\le x\le9$ to see when the remainder is between $0$ and $9$.

 It's only possible for $x=2$. Thus $E=2$ and $N=8$. Then $400O+312=100T+12$, $O=1$ and $T=7$.

A: 'ONE' represents the number $100 \times O + 10 \times N + E$, just as $781$ represents $7 \times 100 + 8 \times 10 + 1$. This is what the decimal position
system is. 
So 'ONE' added 4 times to itself is just $400 \times O + 40 \times N + 4E$ and this should represent the same number as 'TEN' = $100 \times T + 10\times E + N$.
This means 'N' is a the final digit of a multiple of $4$, so $2,4,6,8,0$ are the options for 'N'. For more hints, see the comments.   
