Does 35 divided by 7 mean 7 groups of 5 or 5 groups of 7? I define a group as a set. A set contains members. Moreover, a set is a logical container that I intend to manipulate. The phrase "35 divided by 7", to me, implies there are 7 sets. It is the similar to divide the inheritance among 7 heirs.
37 % 7 gives 2 meaning 5 groups of 7 with 2 remaining in the last group. Based on the definition above, there are 7 sets. The modulus is "2"; "2" is a not a set. If you evenly divide the inheritance among 7 heirs so each heir gets an equal share, then the remaining inheritance is the modulus.
However, if we cover 35 metres in 7 seconds it means we covered 5 metres in 1 second and there are 7 such seconds ie 7 groups of 5. Here we are talking about a rate specifically velocity. Rates are not the same as sets.
Unless you are talking about abstract math, these concepts should tie into the real world. A set is a logical division of something. I have 35 apples that I want to evenly divide among 7 students (7 sets). Thus each student will get 5 apples (5 members). Because I am the world's foremost glutton, I can devour 5 apples every second (rate).
You can also say we found 5 groups of 7, but this phrasing seems to emphasize something different in the context of the original problem. If you say "5 groups of 7", then the group is "the number of people we need in order to allocate 5 apples evenly." This sounds confusing and there's probably a better way to write it, but I think you get the point.
So, sure, 5x7 = 7x5. That's a fact of arithmetic and, if we're just talking about logic and numbers, we can be a little careless. But if you're representing something with the numbers, it's a little different.
Where is this coming from? The modulus is just integer division with a remainder. If you use your theory of interpreting this you can do it both ways.
Division is taking something and spliting it in X parts. By definition: the act of separating something into parts or groups
So when you are divinding by X you are separating the initial quantity into X equal parts.
35/7 means divide 35 in 7 equal parts.
This means each part has 5 things in it and there are 7 such parts which is fundamentally irreconcilable with the modulus operator. I think it is because of the limitation of the modulus operator which only works for integer values. So what you’re saying makes sense with real numbers BUT the context changes when you’re working with integers. So, with integer values it is 5 groups of 7 but 7 groups of 5 with real numbers
Modulus in this case means divide into 7 parts and when you cannot divide anymore keep the reminder.
If x = a*b+c
x ℅ a = c and x % b = c
41 % 7 = 6
41 % 5 = 1