Solve the equation

2007-09-08 11:48 pm

Those creepy Meepits are at it again, coming up with math problems for you to solve!
In the following equation, M, E, and P represent specific digits.
(MEEP)base 5 + (MEEP)base 7 = (MEEP)base 8
What is the number represented by MEEP? For example, if M = 7, E = 8, and P = 9, you would submit the answer "7889"

回答 (1)

abaform

2007-09-09 9:30 am

✔ 最佳答案

In base b, the value of MEEP = P + E * b + E * b^2 + M * b^3
Since (MEEP)base 5 + (MEEP)base 7 = (MEEP)base 8,
(P + 5E + 25E + 125M) + (P + 7E + 49E + 343M) = P + 8E + 64E + 512M
2P + 86E + 468M = P + 72E + 512M
P + 14E = 44M
As MEEP can be expressed in base 5, M, E and P are digits in {0, 1, 2, 3, 4}
A trivial answer will be (M, E, P) = (0, 0, 0) and MEEP = 0000
We can retrieve other answers by observation and some trial-and-error tests.
Since P = 44M - 14E = 2(22M - 7E), P must be an even integer, i.e. {0, 2, 4}.
If P = 0, 22M = 7E, no integral (M, E) pairs satisfies the condition.
If P = 2, 22M - 7E = 1, (M, E) = (1, 3), MEEP = 1332
If P = 4, 22M - 7E = 2, (M, E) = (2, 6), MEEP = 2664, (E out of range)
So, MEEP can be 0000, 1332.

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