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

Let a = 3 + f, where 0 < f < 1. We are given that (3 + f)(3 - 2f) is an integer. This implies that 2f2 + 3f is an integer. Since 0 < f < 1, we have 0 < 2f2 + 3f < 5.

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

Therefore a takes the values: