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       Recall that a composite number is a nonprime positive integer. We will call a composite number n a simple composite if it is the product of distinct primes and a complex composite if it has a factor of the form p² for some prime p.
       Write 12,167 in the form

where each a_i is a complex composite and b is a simple composite.

       The Problem Solving Competition uses problems submitted by professors and students from different campuses throughout the United States and abroad. Each month the competition sends out problems to participating institutions. 
       To submit your solutions to the current problems, give them to Cathie Trigueiro in 108 Deady. Your solutions should be clearly written and should show all of your work and contain some mathematics. A correct solution for each problem will qualify for a drawing. You don't have to solve all of the problems. Just submit the ones you can solve. The person with the winning draw will receive a ten dollar gift certificate to Pegasus Pizza. Of course, this competition is for undergrads only.

provided by the Assistant Editor

             Suppose that A is an nxn matrix with the property that A³ = I, the identity matrix.   Prove that the trace of
A + A² + I is zero.

We received solutions from No One for the puzzles from the last issue. So the winner of the drawing for the pizza for the second straight time is No One! Congratulations!