Loan and credit

I wanted owner builder mortgage. This was just a general inquiry over the phone with no credit check. The lady on the other side of the phone told me that I would need a score of 720-740 to qualify, and that the score would be the biggest factor. I presently have 657, 659, and 700. I am about to make the last payment on my car loan. All payments have been made on time. I also have had a credit card for about 3 months. That is all that I am currently paying on. Depending on which report you look at, I have a few other items in good standing that were paid off a few years ago. The question I had was: Is it possible to achieve a 720 or higher score without having a mortgage on your credit report. If so, what would be the best way for me to proceed?