Question

In a seminar, the number of participants in German, English and French are 204, 120 and 168 respectively. Find the numbers of rooms required to house them if in each room, the same number of participants are to be accommodated and all of them must belong to the same language.

Answer 1

$\bf \stackrel{German}{204}\qquad \stackrel{English}{120}\qquad \stackrel{French}{168}$

now, if we a "prime factoring" of each number, we get

204 = 2*2*3*17

120 = 2*2*2*3*5

168 = 2*2*2*3*7

now, notice the GCD for all three is that common bolded values, namely 2*2*3, or 12, so each room will house 12 participants,

now, how many rooms for all German?  204 ÷ 12, or 17 rooms.

how many rooms for all English ones?  120 ÷ 12, or 10 rooms.

how many rooms for the French ones?  168 ÷ 12, or 14 rooms.

so the total amount of rooms for all of them is 17 + 10 + 14.