Let `f(x)=A x^2+B x+c ,w h e r eA ,B ,C` are real numbers. Prove that if `f(x)` is an integer whenever `x` is an integer, then the numbers `2A ,A+B ,a n dC` are all integer. Conversely, prove that if the number `2A ,A+B ,a n dC` are all integers, then `f(x)` is an integer whenever `x` is integer.

Explanation

Suppose `f(x)=Ax^(2)+Bx+C` is an iteger, whenever x is an integer.
`therefore f(0), f(1),f(-1)` are integers.
`implies C, A+ B+C,A-B` are integers.
`implies C, A+ B,A` are integers.
`implies C, A+B,(A+B)-(A-B)=2A` are integers.
Conversely, suppose `2A,A+B and C` are integers.
Len n be any integer. We have,
`f(n)=An^(2)+Bn+C=2A[(n(n-1))/(2)]+(A+B)n+C`
Since, n is an integer, `n(n-1)1//2` is an integer. Aslo, `2A,A+B and C` are integers.
We get f (n) is an integer for all integer n.