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Find all real numbers a such that 4 < a < 5 and a(a – 3{a}) is an integer (Here {a} denotes the fractional part of a. For example {1.5} = 0.5 ; {–3.4} = 0.6.

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4 < a < 5

Let a = 4 + f

a (a – 3 {a}) = I

(4 + f) (4 + f – 3f) = I

(4 + f) (4–2f) = I

16 – 4f – 2f2 = I

2f2 + 4f = I’

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