LCM and HCF
 Four runners started running simultaneously from a point on a circular track. They took 200 seconds, 300 seconds, 360 seconds and 450 seconds to complete one round. After how much time do they meet at the starting point for the first time ?

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Required time = LCM of 200, 300, 360 and 450 seconds
We find , LCM of 200, 300, 360 and 450 seconds = 1800 secondsCorrect Option: A
Required time = LCM of 200, 300, 360 and 450 seconds
We find , LCM of 200, 300, 360 and 450 seconds = 1800 seconds
∴ Required time = 1800 seconds
 From a point on a circular track 5 km long A, B and C started running in the same direction at the same time with speed of 2^{1}/_{2} km per hour, 3 km per hour and 2 km per hour respectively. Then on the starting point all three will meet again after

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A makes one complete round of the circular track in
5 = 2 hours, 5 2 B in 5 hours and C in 5 hours. 3 2
That is after 2 hours A is at the starting point,B after 5 hours and C after 5 hours. 3 2 Hence the required time = LCM of 2, 5 and 5 hours 3 2 Required time = LCM of 2, 5, 5 HCF of 1 , 3 , 2
Correct Option: C
A makes one complete round of the circular track in
5 = 2 hours, 5 2 B in 5 hours and C in 5 hours. 3 2
That is after 2 hours A is at the starting point,B after 5 hours and C after 5 hours. 3 2 Hence the required time = LCM of 2, 5 and 5 hours 3 2 Required time = LCM of 2, 5, 5 HCF of 1 , 3 , 2 Required time = 10 = 10 hours. 1
 The traffic lights at three different road crossings change after 24 seconds, 36 seconds and 54 seconds respectively. If they all change simultaneously at 10 : 15 : 00 AM, then at what time will they again change simultaneously?

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LCM of 24, 36 and 54 seconds = 216 seconds = 3 minutes 36 seconds
∴ Required time = 10 : 15 : 00 +
3 minutes 36 secondsCorrect Option: B
LCM of 24, 36 and 54 seconds = 216 seconds = 3 minutes 36 seconds
∴ Required time = 10 : 15 : 00 +
3 minutes 36 seconds
Hence , Required time = 10 : 18 : 36 a.m.
 Four bells ring at intervals of 4, 6, 8 and 14 seconds. They start ringing simultaneously at 12.00 O’clock. At what time will they again ring simultaneously ?

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LCM of 4, 6, 8, 14 = 168 seconds = 2 minutes 48 seconds
They ring again at 12 + 2 min. 48 sec.Correct Option: A
LCM of 4, 6, 8, 14 = 168 seconds = 2 minutes 48 seconds
They ring again at 12 + 2 min. 48 sec.
Required time = 12 hrs. 2 min. 48 sec.
 When a number is divided by 15, 20 or 35, each time the remainder is 8. Then the smallest number is

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As we know that when a number is divided by a, b or c leaving remainders p, q and r respectively such that the difference between divisor and remainder in each case is same i.e., (a – p) = (b – q) = (c – r) = t (say) then that (least) number must be in the form of (k – t), where k is LCM of a , b and c .
LCM of 15, 20 and 35 ( k ) = 420
Here , remainder = 8Correct Option: A
As we know that when a number is divided by a, b or c leaving remainders p, q and r respectively such that the difference between divisor and remainder in each case is same i.e., (a – p) = (b – q) = (c – r) = t (say) then that (least) number must be in the form of (k – t), where k is LCM of a , b and c .
LCM of 15, 20 and 35 ( k ) = 420
Here , remainder = 8
∴ Required least number = 420 + 8 = 428