A =72 seconds, B =80 seconds and C =90 seconds.
Their least common multiple is 720 seconds (or 12 minutes), so they meet at the starting point after 720 seconds.
2) Each floor of the two buildings is 12. The new building is 2 meters 80 centimeters high, and the old building is 3 meters 20 centimeters high. On which floor is the ceiling of the two floors flush?
Height of new building = 280cm; height of old building = 320cm.
The least common multiple of 280 and 320 is 5600. So at the height of 5600 cm, the ceiling is flush.
5600/280=8 5600/320=7
The ceilings on the eighth floor of the new building are flush with those on the seventh floor of the old building.
(3) There is a natural number, divided by 10 to 7, divided by 7 to 4, and divided by 4 to 1. What is the smallest natural number?
This number plus 3 is a multiple of 10, plus 3 is a multiple of 7, and plus 3 is a multiple of 4.
The least common multiple of10,7,4 is 280.
So the answer is 280-3=277.
(4) There are 37 telephone poles on the way from school to Children's Palace. The distance between every two adjacent poles used to be 50 meters, but now it has to be changed to 60 meters between every two adjacent poles. Apart from the two poles at both ends, how many poles in the middle don't need to move?
Highway length (37-1) * 50 =1800m.
The least common multiple of 50 and 60 is 300.
1800/300=6 6- 1=5
The answer is: there are five in the middle.
(5) Plant a tree every 4 meters on both sides of the runway. As a result, the distance between the first tree and the last tree is 48 meters. Now transplant trees are planted every 6 meters, and some don't need to be moved.
The least common multiple of 4 and 6 is 12.
48/ 12=4 4+ 1=5 4- 1=3
The answer is: there are three trees in the middle that don't need to be moved. Plus five trees at both ends will not move.