HCF & LCM, General Mathematics

Question: The smallest number which when increased by 5 is divisible by each one of 24, 32, 36 and 54, is :
A
869

B
859

C
4320

D
427

Note: solution: Required number = `(1.c.m. of 24, 32, 36, 54) - 5 = 859`.
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Question: The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18, is :
A
74

B
94

C
184

D
364

Note: solution: L.C.M. of `6, 9, 15, 18 is 90`. Let required number be `90k + 4`, which is a multiple of 7. Letast value of k for which `(90k + 4 )`is divisible by 7 is k=4. `:.` Required number = `90 xx 4 + 4 = 364`.
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Question: The least number which when divided by 18, 27 and 36 leaves the remainders 5, 14 and 23 respectively, is :
A
95

B
113

C
149

D
77

Note: solution: Here `(18 -5 )= 13, (27 -14 ) = 13 & (36 - 23) = 13`. `:.`Required number = `(1.c.m. of 18, 27, 36) - 13 = 95`.
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Question: The least number which when divided by 20, 25, 35 and 40 leaves the remainder 14, 19, 29 and 34 respectively, is :
A
1664

B
1406

C
1404

D
1394

Note: solution: Here `(20 - 14) = 6, (25 - 19) = 6, (35 - 29) = 6 & (40 - 34) = 6`. `:.` Required number = `(1.c.m. of 20, 25, 35, 40)- 6 = 1394`.
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Question: The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is :
A
1677

B
1683

C
2523

D
3363

Note: solution: L.C.M. of `5, 6, 7, 8 = 840`. `:.` Required number is of the from` 840k + 3`. Least value of k for which `(840k +3)` is divisible by 9 is k = 2. `:.` Required number = `(840 xx 2+3) = 1683`.
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Question: The least number which when divided by 5, 6, 8, 9 and 12 leaves a remainder 1 in each case, but when divided by 13 leaves no remainder, is :
A
361

B
721

C
1801

D
3601

Note: solution: L.C.M. of `5, 6, 8, 9, 12 = 360`. `:.` Required number is of the form 360k + 1. Least value of k for which `360k + 1` is divisible by 13 is k = 10. `:.` Required number = `(360 xx 10 +1) = 3601`.
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Question: An electronic device makes a beep after every 60 sec. Another device makes a beep after every 62 sec. They beeped together at 10 a.m. The time when they will next make a beep together at the earliest, is :
A
10.30 a.m.

B
10.31 a.m.

C
10.59 a.m.

D
11 a.m.

Note: solution: L.C.M. of 60 and 62 seconds is 1860 sec = 31 min. `:.` They will beep together at 10.31 a.m.
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Question: Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 second respectively. In 30 minutes, how many times do they toll together ?
A
4

B
10

C
15

D
16

Note: solution: L.C.M. of 2, 4, 6, 8, 10, 12 is 120. So, the bells will toll together after every 120 seconds i.e. 2 minutes. In 30 minutes, they will toll together in `(30/2)` + 1 =16 times.
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Question: The traffic lights at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at 8 :20 :00 hours, then they will again change simultaneously at :
A
8 :27 : 12 hrs

B
8 : 27 : 24 hrs

C
8 : 27 : 36 hrs

D
8 : 27 : 48 hrs

Note: solution: Interval of change = `(1.c.m. of 48, 72, 108) sec. = 432` sec. `:.`The lights will change simultaneously after every 432 seconds, i.e. 7 min. 12 sec. `:.` Next simultaneous change will take place at` 8 : 27 : 12` hrs.
1. Report

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Question: The smallest number which when increased by 5 is divisible by each one of 24, 32, 36 and 54, is :

Question: The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18, is :

Question: The least number which when divided by 18, 27 and 36 leaves the remainders 5, 14 and 23 respectively, is :

Question: The least number which when divided by 20, 25, 35 and 40 leaves the remainder 14, 19, 29 and 34 respectively, is :

Question: The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when divided by 9 leaves no remainder, is :

Question: The least number which when divided by 5, 6, 8, 9 and 12 leaves a remainder 1 in each case, but when divided by 13 leaves no remainder, is :

Question: An electronic device makes a beep after every 60 sec. Another device makes a beep after every 62 sec. They beeped together at 10 a.m. The time when they will next make a beep together at the earliest, is :

Question: Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 second respectively. In 30 minutes, how many times do they toll together ?

Question: The traffic lights at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at 8 :20 :00 hours, then they will again change simultaneously at :