You are correct in your thinking - and I think some of my students solve mod problems exactly like this. When the numbers are small, it works pretty well. When they get larger its a little harder to do in my head and I resort to division. To make the division a little clearer here are some thoughts:
I think the thing that the thing that made mod click for me when dividing, is that I’m trying to divide evenly. When I teach this to students, I tell them to remember back to a time in their lives (3rd, 4th grade maybe) when they learned to divide and did NOT know what a decimal is. When they learned this math, they simply figured out how many times A went evenly in to B, and everything leftover was just that - leftover. We did not subdivide it. It was a ‘remainder’. For example:
11 % 2:
11 evenly goes in to 2 five times. What is left over? 1.
Visually, distributing things to people is always helpful:
I have 11 dice, I have to be fair so each person gets the same number of dice. If I’m dividing between 2 people, we each get 5 dice. That would be 11/2 = 5. BUT when we talk about mod (%) we don’t actually care about 5, we care about how many dice were left over after the even distribution has occurred, that’s the one left over die. So 11 % 2 is 1.
Here’s another forum post that also explains mod through long division, and links to a khan academy video that makes the connection between mod and ‘clock arithmetic’. I hope that one of these options helps clear it up!