Let f(x) = Ax 2 + Bx + C, where A, B, C are real numbers. Prove that if f(x) is an integer whenever x is integer, then the numbers 2A, A + B and C are all integers. Conversely, prove that if the numbers 2A, A + B and C are all integer then f(x) is an integer whenever x is an integer.
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(i) f(x) = Ax 2 + Bx + C ⇒ x ∈ Ι and f(x) ∈ Ι
at x = 0, f(0) = C ⇒ C is integer at x = 1, f(1) = A + B + C
C is integer ∴ A + B is also integer
at x = –1, f(–1) = A – B + C ⇒ f(1) + f(–1) = 2A + 2C
C is integer ∴ 2A is also integer
(ii) f(x) = A x(x – 1) + (A + B) x + C ⇒ f(x) = 2A 
+ (A + B)x + C
If x is an integer then 
is also an integer and 2A, 
(A+ B), C ∈ Ι
⇒ f(x) is also an integer.
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