# H.C.F and L.C.M

Q11. If two numbers are in the ratio 7:5 and their L.C.M is 315, then their sum is

(a) 108

(b) 123

(c) 135

(d) 141

**Answer:** (a) 108

Let the numbers be 7x, 5x

∴ L.C.M = 35x

A/Q 35x = 315

⇒ x = 315/35 = 9

∴ The numbers are 63, 45

∴ Sum = 63 + 45 = 108

Q12. The least perfect square number which is divisible by 4, 5, 6 and 8 is

(a) 3200

(b) 3400

(c) 3600

(d) 3800

**Answer:** (c) 3600

2|4, 5, 6, 8 |

2|1, 5, 3, 4 |

1, 5, 3, 2 |

L.C.M = 2 X 2 X 5 X 3 X 2

= 2^{2} X 5 X 3 X 2

∴ Required Number = 2^{2} X 5^{2} X 3^{2} X 2^{2}

= 4 X 25 X 9 X 4 = 100 X 36 = 3600

Q13. The smallest number which is divisible by 12, 15, 25 and is a perfect square is

(a) 3100

(b) 3350

(c) 3500

(d) 3600

**Answer:** (d) 3600

3|12, 15, 25 |

5|4, 5, 25 |

4, 1, 5 |

L.C.M = 3 X 5 X 4 X 5

= 3 X 5^{2} X 4

∴ Required Number = 3^{2} X 5^{2} X 4^{2}

= 9 X 25 X 16 = 3600

Q14. The traffic lights at three different road crossing change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8.20 hours, then at what time will they again change simultaneously ?

(a) 8.22.15 hrs

(b) 8.27.12 hrs

(c) 8.28.35 hrs

(d) 8.30.30 hrs

**Answer:** (b) 8.27.12 hrs

2|48, 72, 108 |

2|24, 36, 54 |

3|12, 18, 27 |

3|4, 6, 9 |

2|4, 2, 3 |

2, 1, 3 |

L.C.M of 48, 72, 108 = 2 X 2 X 3 X 3 X 2 X 2 X 3 = 432

∴ Intervel of change = 432 seconds = 7 min 12 sec

∴ Time of next change = 8 : 27 : 12 hrs

Q15. 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 next time, when they would beep together at the earliest

(a) 10.31 am

(b) 10.39 am

(c) 10.46 am

(d) 10.51 am

**Answer:** (a) 10.31 am

2|60, 62 |

30, 31 |

L.C.M of 60, 62 = 2 X 30 X 31 = 60 X 31 = 1860

∴ Interval of beeping together = 1860/60 min = 31 min

∴ They will beep together again at 10.31 am

Q16. What is the least number which when divided by 9, 12, 15 leaves a remainder 2 in each case ?

(a) 182

(b) 189

(c) 197

(d) 201

**Answer:** (a) 182

3|9, 12, 15 |

3, 4, 5 |

L.C.M of 9, 12, 15 = 3 X 3 X 4 X 5 = 180

∴ Required number = 180 + 2 = 182

Q17. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 seconds respectively. In 30 minutes, how many times do they toll together ?

(a) 16 times

(b) 19 times

(c) 21 times

(d) 24 times

**Answer:** (a) 16 times

L.C.M of 2, 4, 6, 8, 10, 12 = 2 X 2 X 2 X 5 X 3 = 120 seconds = 2 mins

∴ After every 2 minutes they toll together

Number of times they toll in 30 minutes = 30/2 + 1 = 16 times

Q18. The L.C.M of two prime numbers a and b (a > b) is 161. The value of 2a - b is

(a) 29

(b) 32

(c) 39

(d) 43

**Answer:** (c) 39

161 = 7 X 23

∴ a = 23 , b = 7

∴ 2a - b = 2 X 23 - 7 = 46 - 7 = 39

Q19. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 seconds respectively. In 28 minutes, how many times do they toll together.

(a) 11 times

(b) 12 times

(c) 15 times

(d) 19 times

**Answer:** (c) 15 times

L.C.M of 2, 4, 6, 8, 10, 12 = 2 X 2 X 2 X 5 X 3 = 120

After every 120 sec or 2 minutes they toll together

∴ Required number of times = 28/2 + 1 = 14 + 1 = 15 times

Q20. The ratio between two numbers is 4:5. If their L.C.M is 180, then the numbers are

(a) 14, 24

(b) 16, 38

(c) 30, 42

(d) 36, 45

**Answer:** (d) 36, 45

Let, the numbers be 4x, 5x

∴ L.C.M = 4 X 5 X x = 20x

A/Q 20x = 180 ⇒ x = 180/20 = 9

∴ the numbers are 36, 45