I dont know if there is algebraic way of solving this faster but I just worked this out by substitution (speedy since the number of employees is small)

10 people in june - so even we assume 9 in all other months
10 + 9*11 = 109 (less than total employees)
similarly for 11 in june
11 + 10*11 = 121 less than total employees
for 12 in june
12 + 11 * 11 = 133 less than total employees

So 13 fits the bill and hence would be the least possible for June.

I would be interested to see other simpler and better solutions ..anybody?

135/12 = 11 with remainder of 3. Therefore, you have 11 people who are born in each month with 3 to distribute to each month - attempting to minimize June but keep it as maximum, at the same time.

You can give 1 person to, say, September, which will make its output 12. You then must keep 2 for June, which will give it 13 people.

Hey, I was born in June, I guess my month wins.