Residuals uncorrelated with predictor variables assumption

I'm studying the review for one of my graduate classes, Correlation and Regression Analysis, and this is one of the questions:

"A friend says that you don't have to worry about the assumption that the residuals are uncorrelated with each predictor variable: each time she has conducted a regression analysis, the residuals always turn out to be uncorrelated with the predictors by mathematical necessity. Explain for your friend (a) what is really meant by this assumption, (b) why your friend came to the wrong conclusion trying to detect violations of the assumption in the manner described, and (b) the consequences of violating the assumption."

I'd really appreciate some help with it.