When the number 777 is divided by the integer N. . . .

Upon reading this problem, we know that the factor must be greater than 77 to give that as a remainder. We also know that the number must divide 700 evenly. So how many numbers are there that divide 700 evenly and that are greater than 77?

700 can be prime factored to 10*10*7, which further reduces to 1*2*2*5*5*7. With these numbers in mind, we can determine how many factors there are:

700 / 1 = 700 is one possible factor.
700 / 2 = 350 is another.
700 / 2 /2 = 175 is another.
700 / 5 = 140 is another factor.
700 / 7 = 100 is the final factor.

So there are 5 possible factors. That's answer choice (D)