vaibhav101 wrote:There are 315, 420 and 1155 boys of three different schools taking part in an essay competition. What is the minimum number of rooms required such that all the rooms have equal number of students and each room has students of the same school only?
A 12
B 14
C 16
D 18
E 20
To MINIMIZE the number of rooms, we must MAXIMIZE the number of students per room.
Let x = the number of students per room.
Since each room must have x students -- and we want x to be as large as possible -- x must be equal to the GREATEST COMMON FACTOR of 315, 420, and 1155:
315 = 3*
3*5*7
420 = 2*2*
3*5*7
1155 =
3*5*7*11.
As illustrated by the blue values above, the GCF of 315, 420 and 1155 =
3*5*7.
Since x = 3*5*7, we get:
Number of rooms required for the first school = (total number of students)/(x students per room) = (3*3*5*7)/(3*5*7) = 3.
Number of rooms required for the second school = (total number of students)/(x students per room) = (2*2*3*5*7)/(3*5*7) = 4.
Number of rooms required for the third school = (total number of students)/(x students per room) = (3*5*7*11)/(3*5*7) = 11.
Total number of rooms = 3+4+11 = 18.
The correct answer is
D.
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