A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the stud

Question

A school has less than 5000 students and if the students are divided equally into teams of either 9 or 10 or 12 or 25 each, exactly 4 are always left out. However, if they are divided into teams of 11 each, no one is left out. The maximum number of teams of 12 each that can be formed out of the students in the school is

Accepted Answer

150 — Easy "Leaves remainder $4$ when divided by $9,10,12,25$" means the count is $	ext{LCM}+4$ form. Impose the two extra conditions — under $5000$ and divisible by $11$ — to pin the exact number, then divide by $12$. 1 Translate the common remainder. The number $N$ satisfies $N\equiv 4$ for all of $9,10,12,25$: $$N=	ext{LCM}(9,10,12,25)\,k+4=900k+4$$ 2 Apply $N Test $k=1,2,3,4,5$: $$\begin{aligned} &904,\ 1804,\ 2704,\ 3604,\ 4504\ &1804=11	imes 164\ \checkmark \quad(	ext{the only multiple of }

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