Mathematics Problem Of the Week

Fall 2007

POW #3

Basic Valences

We know that in our usual (base 10) number system we can easily tell if a positive integer is odd or even by just looking at the valence (odd or even) of the last digit. Does this work in other bases?

Examples:

143₆ = 1*6² + 4*6 + 3 = 63 is odd

202₃ = 2*3² + 0*3 + 2 = 20 is even

1231₅ = 1*5³ + 2*5² + 3*5 + 1 = 191 is odd

(a) Prove that, if the base is even, the usual rule (whether or not the last digit is odd or even) still works.

(b) Does the same rule also work for odd bases? If so, prove it. If not, find (and justify) an easy way to determine if a number expressed with an odd base is even.

Due Friday, September 21^st at Noon.

Solutions should be submitted to Dr. R. Lock's mailbox in the Math/CS/Stat office or sent via e-mail to rlock@stlawu.edu.