SOLUTION: If a + b are odd integers, which of the following must be an even integer? I. {{{(a + b)/2}}} II. {{{ab - 1}}} III. {{{(ab + 1)/2}}} Possible answers: a) I only b) II

Algebra ->  Equations -> SOLUTION: If a + b are odd integers, which of the following must be an even integer? I. {{{(a + b)/2}}} II. {{{ab - 1}}} III. {{{(ab + 1)/2}}} Possible answers: a) I only b) II       Log On


   

Question 479594: If a + b are odd integers, which of the following must be an even integer?
I. %28a+%2B+b%29%2F2
II. ab+-+1
III. %28ab+%2B+1%29%2F2
Possible answers:
a) I only
b) II only
c) I and II only
d) II and III only
e) I, II and III
I would appreciate your help. Thanks! :-)
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!

  I. %28a+%2B+b%29%2F2 NO, because %283%2B7%29%2F2 = 10%2F2 = 5, which is ODD  

 II. ab+-+1 YES, because an ODD times an ODD is an ODD, 
so ab is always ODD, and then when we subtract 1 from that ODD, 
we will always get an EVEN.

or algebraically,

(2m+1)(2n+1)-1 = 4mn+2m+2n+1-1 = 2(2mn+m+n) = which is EVEN.

III. %28ab+%2B+1%29%2F2  NO, because %283x7%2B1%29%2F2=%2821%2B1%29%2F2=22%2F2 = 11, which is ODD

Therefore only II is always even.  Answer b. 

Edwin