Answer to Question #240274 in Combinatorics | Number Theory for 0.0

Question

There are 18 boys and 35 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

$n(B)= 18\\ n(G)=35\\ \text{If the two groups must get the same number of chips,}// \text{the k is an even number }\\ \implies n(B_c)= n(G_c) =\frac{k}{2}\\ \text{If each pupil get at least one then}\\ \frac{k}{2-n(B)} \ge 1 \text{ and } \frac{k}{2-n(G)} \ge 1\\ \text{that is} \frac{k}{36} \ge 1 \text{ and } \frac{k}{70} \ge 1\\ \text{Since LCM}(36,70) = 1260,\\ \text{hence, we have that the highest possible value of k is } 1260$