Vignettes #57

The Munich newspaper Süddeutsche Zeitung writes out annually a summer quiz which is notorious for its difficult riddles, over 4 weeks of about 40 brainteasers. At the end it's merely amazing that there is any winner at all. In the early 90s there was the following question:
A customer at the customer buys 4 items and goes with them to the cash register. The cashier types in the prices and says: "That makes $7.77 altogether." The customer protests: "I saw that you multiplied the 4 prices, but you must add them!" The cashier makes an apology and this time adds the prices up. "Well surprise: the sum is also $ 7.77!" So, the question is what are the prices of the 4 items which you can add up or multiply and still the result will be $7.77.
My pal Gerhard is a mathematician, and he told me that at our university all mathematicians working at his faculty were trying to figure out this problem with 4 unknowns. It is not solvable with a simple equation. Actually these experts agreed that you can only find the solution by going through all possible multiplications. So they wrote a program and fed it to the university computer. This was in the early 90s, so it took the computer all night to solve the problem, I guess it ran for 8-10 hours.

I argued that this can't be the way to solve a riddle that is directed to a general audience albeit a more intellectual one. There must be some way to solve it without using highly sophisticated equipment. Basically you'd need to simply try out some variations using simple assumptions, but my pal Gerhard is convinced the only stringent approach is to go through all factors...
After I found a small hint I did get the solution, but Gerhard still disagrees concerning the approach.