Remainders Question from MGMAT

RoseSignet wrote:I'm not sure where to begin with this question.

more to the point, here's a helpful TECHNIQUE for beginning problems like this one.

if you have a problem about REMAINDERS, you should view that problem as an opportunity for PATTERN RECOGNITION.
there are lots of topics that lead to recognition of common patterns - i.e., remainders aren't the only topic of such problems - but, in remainder problems, CLEAR patterns tend to emerge QUICKLY if you start testing numbers in some sort of systematic manner.

in this problem, therefore, and in problems like it:
if you don't immediately see a better technique, you should JUST START PLUGGING IN SAMPLE NUMBERS AND LOOK FOR A PATTERN.

let's do this with the first several of "the integers between 10 and 110, inclusive" (the set in question), and see what happens:
10 --> remainder = 3
11 --> 4
12 --> 5
13 --> 6
14 --> 0
15 --> 1
16 --> 2
17 --> 3
18 --> 4
19 --> 5
20 --> 6
21 --> 0
by this point it's pretty clear what's happening: the remainders go 0, 1, 2, 3, 4, 5, 6, and then repeat the same sequence over and over again.

the posters above have the correct answer nailed down already, but note how much easier it is to resolve (b) once you've found this pattern.