There is a bar of length 180cm.
I mark points on the bar where it is divided M equally and N equally respectively.
M and N are integers and M is bigger than 2 and is smaller than N.
For example when M is 4 and N is 6, it is marked respectively.
As the mark at one point is overlapped as shown in the lower figure, number of point becomes seven.
Answer the following questions.
(1) In case M is 5 and N is 6, find the number of point.
Find the shortest length of interval between two points.
(2) In case M is 8 and N is 12, find the number of point.
Find the shortest length of interval between two points.
(3) There are two cases of combination of M and N which make the number of point eight.
Answer these combinations of M and N.
(4) Answer all combinations of M and N that the shortest length becomes 12cm among the length of the interval between points.
Answer
(1) 9 pieces, 6 cm
(2) 15 pieces, 7.5 cm
(3) (3,7),(3,9)
(4) (3,5),(3,15),(5,15)
Solution
(1) 180 / 5 = 36 (cm) 0 36 72 108 144 180
180 / 6 = 30 (cm) 0 30 60 90 120 150 180
Number of mark is nine.
The shortest is 36 - 30 = 6 cm.
(2) 180 / 8 = 22.5 (cm) 0 22.5 45 67.5 90 112.5 135 157.5 180
180 / 12 =15(cm) 0 15 30 45 60 75 90 105 120 135 150 165 180
Number of mark is 15.
The shortest is 22.5 - 15 = 7.5 cm.
(3)
In case M is 3 and N is 7, there is no mark overlapped.
Thus number of mark is 8.
In case M is 3 and N is 9, there are two overlapping points.
Thus number of mark is 2 + 8 -2 = 8.
(4) According to the table below, (M,N) = (3,5),(3,15),(5,15).
In case of (3,5), 36 × 2 -60 = 12.
In case of (3,15), 60- 12 × 4 = 12.
In case of (5,15), 36 - 12 × 2 = 12.
