If a, b, and c are integers and a(b - c) is odd, which of the followin

Here Odd = o and Even =e
a………..b………..c……….a(b-c)
o………..o……….e……….o*(o-e) =o……..First case
o………..e……….o……….o*(e-o) = o…….Second Case

From the cases above, ab +c = o + e = o
ab + c = e + o = o
Ans:D

Question:
If a, b, and c are integers and a(b - c) is odd, which of the following must be odd?
A. b
B. c
C. ab
D. ab + c
E. abc


Solution:

a(b - c) is odd
=> a is odd AND (b - c) is odd
=> a is odd AND b is even and c is odd ... (possibility 1)
OR a is odd AND b is odd and c is even ... (possibility 2)

Let us check the options:
A) b : As discussed above, b could be even or odd - hence, incorrect
B) c : As discussed above, b could be even or odd - hence, incorrect
C) ab: As discussed above, b could be even or odd => hence, ab could be even or odd - hence, incorrect
D) ab + c:
According to Possibility (i): odd x even + odd = odd
According to Possibility (ii): odd x odd + even = odd
Hence, this option is correct
E) abc: It is clear that since b or c may be even or odd, abc could also be even or odd - hence, incorrect

Answer D

The only way to obtain an odd product is for both of the factors to be odd, since odd x odd = odd.

Since a(b - c) is odd, then a must be odd and (b - c) must be odd, which means b is odd when c is even, or vice versa. Thus, ab will be even when c is odd, or ab will be odd when c is even. In any case, ab + c is odd.

Answer: D

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