# Find all real numbers a such that 4 < a < 5 and a(a − 3{a}) is an integer.

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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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Let a = 4 + f, where 0 < f < 1. We are given that (4 + f)(4 − 2f) is an integer.

This implies that 2f2 + 4f is an integer. Since 0 < f < 1, we have 0 < 2f2 + 4f < 6.

Therefore 2f2 + 4f can take 1, 2, 3, 4 or 5. Equating 2f2 + 4f to each one of them and using f > 0, we get