February 2010 Brain Teaser answer and solution

Solve for a, b, c, d, and e.

a (b + c + d + e) = 128
b (a + c + d + e) = 155
c (a + b + d + e) = 203
d (a + b + c + e) = 243
e (a + b + c + d) = 275

Check out the prime factors of each of the numbers.
155 = 5 x 31...therefore b = 5
203 = 7 x 29...therefore c = 7
243 = 3 x 81...therefore d = 3
275 = 11 x 25...therefore e = 11

a (b + c + d + e) = 128
a(5 + 7 + 3 + 11) = hey this doesn't work

ok let's try 243 = 9 x 27 so d = 9
a(5 + 7 + 9 + 11) = a ( 32) therefore a = 4
4(5 + 7 + 9 + 11) = 128

Check:
5(4 + 7 + 9 + 11) = 5(31) = 155
7(4 + 5 + 9 + 11) = 7(29) = 203

9(4 + 5 + 7 + 11) = 9(27) = 243

11(4 + 5 + 7 + 9) = 11(25) = 275

so: a = 4,b = 5, c = 7, d = 3, e = 11