factors

Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is non-negative integer and always less than divisor).

So we'd have: \(777=qn+77\), where \(remainder=77<n=divisor\) --> \(qn=700=2^2*5^2*7\) --> as \(n\) must be more than 77 then \(n\) could take only 5 values: 100, 140, 175, 350, and 700 (the least factor more than 77 is \(2^2*5^2=100\), now if you substitute 2, 2^2, and 5 by 7 you'll get the factors more than 100 plus if you include 7 you'll get one more factor, 700 itself, so total 1+3+1=5).

Answer: D.

Discussed here: remainder-101074.html?hilit=possibilities#p782422

Hope it's clear.