Question

Find all real numbers a such that 3 < a < 4 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).

Answer 1

3 < a < 4

Let a = 3 + f

a (a – 3 {a}) = I

(3 + f) (3 + f – 3f) = I

(3 + f) (3 – 2f) = I

9 – 3f – 2f² = I

2f² + 3f = I’