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

Question 479594: If a + b are odd integers, which of the following must be an even integer?
I.
II.
III.
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) (Show Source): You can put this solution on YOUR website!


  I.  NO, because  =  = 5, which is ODD  

 II.  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.   NO, because  = 11, which is ODD

Therefore only II is always even.  Answer b. 

Edwin

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