for six distinct real solutions of equation mod(x^2-5mod(x)+6)=k we have value of k as
The quadratic function has a minimum value –1/4 (when t=2.5). So the equation f(t) = c has two solutions if c>–1/4, one solution if c=–1/4, and no solutions if c<–1/4. If t=|x| (and t>0), then , so there will be two values of x for each value of t.
Now put that information together to find one value of k for which the equation has six solutions.