1. ## Problem

(PEEM)base 5 + (PEEM)base 7 = (PEEM)base 8

in this equation each of the letters represents a numer.
Like e is 3 and m is 6 and p is 5.
So what number does "Peem" represent?

2. Originally Posted by Horror
(PEEM)base 5 + (PEEM)base 7 = (PEEM)base 8

in this equation each of the letters represents a numer.
Like e is 3 and m is 6 and p is 5.
So what number does "Peem" represent?
The setup is
$PEEM_5 = P \times 5^3 + E \times 5^2 + E \times 5^1 + M \times 5^0 = 125P + 30E + M$

$PEEM_7 = P \times 7^3 + E \times 7^2 + E \times 7^1 + M \times 7^0 = 343P + 56E + M$

$PEEM_8 = P \times 8^3 + E \times 8^2 + E \times 8^1 + M \times 8^0 = 512P + 72E + M$

$(125P + 30E + M) + (343P + 56E + M) = (512P + 72E + M)$

$44P - 14E - M = 0$

Now, we know that P, E, and M are all less than 5 since PEEM is a number in base 5.

So we can start by postulating P = 1:
$14E + M = 44$

M is at most 4, so we need E to be at least 3. (And, as it turns out, less than 4.) So try E = 3:
M = 2.

So PEEM = 1332 works.

I can find no other solutions.

-Dan

3. Since E and M is < 4, 14 E + M is < 15*4 = 60. Any P > 2 would make 44 P > 88, hence the equation doesn't meet in both ends. So P < 1. P = 0 makes 14 E + M = 0, which only solution is E = M = P = 0.

So PEEM is either 1332 as topsquark pointed out or 0000.