Multiplication Question

Question

I have this question for my math class. I've been working on it for about an hour. Here it is:

Each of the letters, F,I,V,E, in this multiplication stands for a different digit. What are the values of the letters?

    FIVE
  X FIVE
  -------
   ****F
  ****I
 ****V
****E
--------
********

From this I've figured out that this essentially means:

E*E % 10 = F, V*E % 10 = I, I*E % 10 = V, F*E % 10 =E

with the % meaning modulo.

I wrote a quick python script to try and brute force it with the code shown below:

tries = 9
for F in range(tries):
    for I in range(tries):
        for V in range(tries):
            for E in range(tries):
                if E * E % 10 == F and V*E % 10 == I and I*E%10 == V and F*V%10 == E:
                    print(F,I,V,E)

and it output:

0 0 0 0
1 1 1 1
5 5 5 5
6 6 4 4
6 6 6 6

Which all fit the criteria but aren't the right answer since there are repeating numbers.

I have no idea how else to approach this. Any help is appreciated.

Answer 1

$F$ is the last digit of a perfect square.

and $F\ne E$

It appears that every based on the asterix that $F>1$ i.e. each line appears to be a 5 digit number and that is not going to happen if the leading digits are $1$'s.

$F\in \{4,6,9\}$

which puts $E\in\{2,3,4,7,8\}$

$VE = I\pmod {10}$ and $IE = V\pmod {10}$

$IE^2 = IF = I\pmod {10}$

$(F = 1)$ or ($F = 6$ and $I,V$ are even).

But we have decided that $F\ne 1$

$F = 6, E = 4$ and $V,I$ are even

$6824$ and $6284$ should both work.

Answer 2

You have a typo in the second to last line of your Python script. Re-run with:

if E * E % 10 == F and V*E % 10 == I and I*E%10 == V and F*E%10 == E:

and the solution (6, 8, 2, 4) shows up.