Equal or Even Distribution/Division

I couldn't think of better subject line - if there's an actual terminology for this, please let me know.

I'm trying to figure out a way to divide a number across several groups of 3, 4, or 'n'.
Say I have a number of people, 37 for example, using groups of 4, that gives me 9 groups and 1 person remaining. Rather than having a single person remaining, I'd like to distribute that more evenly by taking people from other groups. Essentially I'm looking to end with 7 groups of 4 (28), and 3 groups of 3 (9) for a total of 37.

So, 37 into groups of 4 => 7 groups of 4 (28) plus 3 groups of 3 (9) for a total of 37
Or, 42 into groups of 4 => 9 groups of 4 (36) plus 2 groups of 3 (6) for a total of 42.

However, there's one caveat: if the remainder is only ONE away from a full group (from initial division), don't do anything. So taking 35 into groups of 3 as an example, I can have 11 groups of 3 (33), and a single group of 2 for a total of 35. That last group is only 1 away from a full group and that's ok.

So, 35 into groups of 3 => 11 groups of 3, plus 1 group of 2.
But, 34 into groups of 3 => 10 groups of 3, plus 2 groups of 2.

Similarly, in groups of 4:
31 into groups of 4 => 7 groups of 4, plus 1 group of 3.
But, 33 into groups of 4 => 6 groups of 4, plus 3 groups of 3.

If there a way to do this with formulas? Right now I'm manually entering in the information and doing calculations as I fill in groups. See attached Excel file with the sheet I'm currently using to do this manually. I reset everything to 0's then manually start filling in the groups.

Thank you Kaper for providing both solutions, with and without CELING.MATH. They both seem to work fine.

Now I'll go break it up so I can better understand the backend of that formula.

Thanks again!

Probably needs more refining, but in the event that there are a very low number of students, things go wrong. For example, if I were to enter 6 for total students, for column M it shows 2 groups of 4 => that's 2 extra ghost students. If I enter 11 students, column M shows 3 groups of 4. Similarly, if I only enter 5 students, column J shows 2 groups of 3.

12 and higher appears to calculate ok.

To be perfectly honest, no I don't need to know how to split low numbers like that. However, as soon as I shared it with my work collegues, someone immediately spoke up saying she couldn't use small numbers and that the tool is "broken". Despite several of us saying there's no NEED to know how (to split small numbers), I knew the only way to please her (and make her stop complaining) was to have it work properly. She's one of those folks who live by 'Either it works correctly, or it's broken (and doesn't work at all.)"