Answer to Question #247367 in Algebra for M.l

Question

There are 17 boys and 12 girls in a mathematical club. For playing some game, the teacher has to distribute chips among the children (their total number is equal to K, and all of them have to be given). It is necessary that all the boys have the same numbers of chips, all the girls have the same numbers of chips, and each of the children has at least one chip. It has turned out that the teacher can distribute the chips in a single way. Determine the largest possible value of K.

Answer 1

N(B)= 18

N(G)= 35

If the two groups must get the same number of chips, the k is an even number.

=> n(B_i)=n(G_i)= k/2

If each pupil gets a chip, then it follows that

[ k/(2*n(B))] ≥ 1 and [k/(2*n(G))] ≥ 1

That is,

(k/36) ≥ 1 and (k/70) ≥ 1

Since the L.c.m. of (36,70) is then 1260, it follows that the largest possible outcome k can take is 1260.