1. Different letters represent different digits, and the same letter always represents the same digit.
(a) The four-digit number $A D A M$ is the fourth power of some number, and the three-digit number $E V E$ is the cube of some number. Find $A D A M$ and $E V E$.
(b) Adam divides the four-digit number $E D E N$ by $\sqrt{A D A M}$, and Eve divides $E D E N$ by $\sqrt[3]{E V E}$. If the difference between the two remainders is represented by $N$, then, the sum of these two remainders should be represented by which letter? (The letters represent the same digits as in (a) note).
Three, 1. (a) Since $10^{4}=10000, \sqrt[1]{A D A M}$ - must be a $\therefore$ digit number. $5^{4}=625$ is too small, while $6^{4}=1296,7^{4}=2401,8^{4}=$ 4096 do not fit the given form, therefore, $A D A M=9^{4}=$ 6561. Also, since $10^{3}=1000, \sqrt[3]{E V E}$ must also be a one-digit number. $4^{3}=64$ is too small, while $5^{3}=125,6^{3}=216,8^{3}=512,9^{3}=729$ do not fit the given form, therefore, $E V E=7^{3}=343$.
(b) We have already obtained $A=6, D=5, M=1, E=3, V=4$. When $N=0$, $E D E N=3530$, the remainders of the division by Adam and Eve are both 2, their difference is 0, which is indeed $N$, and their sum is also exactly $4=$ $V$. At the same time, we must verify that there are no other possibilities. If $1 \leqslant$ $N \leqslant 4$, the two remainders are still the same, the difference is 0 but not $N$; when 5 $\leqslant N \leqslant 6$, the differences of the remainders are $7-0=7$ and $8-1=7$, which are also not equal to $N$; when $7 \leqslant N \leqslant 9$, the differences of the remainders are sequentially $2-0=2,3$ $-1=2,4-2=2$, which are also not equal to $N$. Therefore, the only possible answer is $V$.