Determine all possible pairs of integers a,b so that exactly one of them is even and so that there are non-integer x,y, such that both x+y and ax+by are integer?
Solution
Since ax+by=a(x+y)+(b−a)y, the value of (b−a)y is integer. If ∣a−b∣=1, then y has to be integer – contradiction.
For ∣a−b∣>1 we can let y=b−a1 and x=−y. Then x+y=0, (b−a)y is integer, hence ax+by is also integer.
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Source: MathNet,
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