A very different Alfred E. Neuman alphametic

Question

Yesterday I posed an Alfred E. Neuman alphametic with multiplication, and today I pose an Alfred E. Neuman alphametic with division for the lovers of Mad magazine:

Every letter and every question mark stands for a digit in base-9 representation. Different letters stand for different digits. Leading digits are always non-zero.

ALFRED $\div$ E = NEUMAN

Which digit does each letter represent? (Please present the full analysis how these digits can be determined.)

Answer 1

I hate base-9.

Lets re-write it as follows:

NEUMAN
x    E
------
ALFRED

OK, so first off, we know $E\notin\{0,1\}$. Since the result is a 6 digit number, we know that $N\in\{1,2,3,4\}$. Since $E \times N$ shows up twice, once with a $D$ result and once with an $A$ result, we know that the fifth column ($E \times E$) must have a carry over.

Also, we will use the fact that the carry over for multiplying any two numbers must be less than the two numbers.

• If $N=4$, then $E=2$ and $A=8$. But then $D=8$ too, which is impossible.
• If $N=3$, then $E=2$ and $D=6 \implies A \in \{7,8\}$. Since $E^2=4$, we need a carry over of at least 4 from the $U \times E$ column, which is impossible.
• If $N=1$, then $E \times N=D \implies E=D$.

Thus $N=2$

So $E=3$ and $D=6 \implies A \in \{7,8\}$. The only value that works for the second column where $A \times E$ yields $E$ is 7 since $7\times 3=23$.

Looking at the fifth column, $E \times E$ yields L. Since $E^2=10$, we know that $L$ is simply the carry over from the previous column, which cannot be greater than 2. Thus, $L\in\{0,1\}$ since 2 is already taken.

Assume, $L=1$

So $U \times E$ has a carry over of 1, $U\in \{4,5\}$.

The remaining values are $\{0,4,5,8\}$. And since $A=7$ the carry over from the second column into the third is 2.

• If $M=0$, then $M \times 3 +2 =2 \implies R=2$, which is already taken.
• If $M=4$, then $M \times 3 +2 =13+2 \implies R=5$, leaving nothing for $U$.
• If $M=8$, then $M \times 3 +2 =26+2 \implies R=8$ too.
• If $M=5$, then $M \times 3 +2 =16+2 \implies R=8$. And $U=4$. This leave $F=0$, but doing the math gets $234572 \times 3=714836 \implies F=4$.

Thus $L=0$

$U \times E$ has a carry over of 0, so $U=1$.

The remaining values are $\{4,5,8\}$.

• $M=8 \implies 231872 \times 3 = 705836$, so $R=8$
• $M=4 \implies 231472 \times 3 = 704536$, so $F=4$

Thus, $M=5 \implies 231572 \times 3 = 704836$ so $R=8$ and $F=4$.

So we have

231572
x    3
------
704836