Problem: Let D=a2+b2+c2 where a,b are consecutive integers and c=a b. Then D​ is:
Answer Choices:
A. always an even integer
B. sometimes an odd integer, sometimes not
C. always an odd integer
D. sometimes rational, sometimes not
E. always irrational
Solution:
D=a2+b2+c2=a2+(a+1)2+(a(a+1))2=a4+2a3+3a2+2a+1=(a2+a+1)2. ∴D​=a2+a+1, an odd integer for any integer a.