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

Q31. Find the H.C.F of (3 X 3^{2} X 5 X 7^{4}), (3^{2} X 3^{3} X 5^{2} X 7) and (3^{3} X 3^{5} X 5^{3} X 7^{5})

(a) 890

(b) 912

(c) 945

(d) 980

**Answer:** (c) 945

H.C.F = 3^{3} X 5 X 7 = 27 X 35 = 945

Q32. Find the H.C.F and L.C.M of 2/3, 8/9, 10/27 and 16/81

(a) 3/70, 71/5

(b) 2/81, 80/3

(c) 5/83, 83/7

(d) 5/99, 92/7

**Answer:** (b) 2/81, 80/3

We have

H.C.F = | H.C.F of 2, 8, 10, 16 | = | 2 |

L.C.M of 3, 9, 27, 81 | 81 |

L.C.M = | L.C.M of 2, 8, 10, 16 | = | 80 |

H.C.F of 3, 9, 27, 81 | 3 |

Q33. Two numbers are in the ratio 6 : 13. If their H.C.F is 5, find the numbers.

(a) 25, 65

(b) 30, 65

(c) 35, 85

(d) 40, 90

**Answer:** (b) 30, 65

Let, the numbers be 6x, 13x

∴ H.C.F = x

Now x = 5

∴ The numbers are 6 X 5, 13 X 5 = 30, 65

Q34. The H.C.F of two numbers is 5 and their L.C.M is 120. If one of the numbers is 40, find the other.

(a) 15

(b) 21

(c) 23

(d) 28

**Answer:** (a) 15

Let, the numbers be x

A/Q L.C.M X H.C.F = 40 X x

⇒ x = | 5 X 120 |

40 |

= 15

Q35. Four electronics devices make a beep after every 20 minutes, 40 minutes, 1 hour and 1½ hour respectively. All the devices beeped together at 12 noon. At what time will they beep together again?

(a) 4 PM

(b) 5 PM

(c) 6 PM

(d) 7 PM

**Answer:** (c) 6 PM

L.C.M of 20 min, 40 min, 60 min, 90 min

2|20, 40, 60, 90 |

5|10, 20, 30, 45 |

2|2, 4, 6, 9 |

3|1, 2, 3, 9 |

1, 2, 1, 3 |

∴ L.C.M = 2 X 5 X 2 X 3 X 2 X 3 = 360 min = 6 hours

So, they will beep together again next at 6 PM.

Q36. Find the least number which when divided by 3, 4, 5, 6, 10 leaves the same remainder 3 in each case.

(a) 63

(b) 65

(c) 69

(d) 72

**Answer:** (a) 63

2|3, 4, 5, 6, 10 |

3|3, 2, 5, 3, 5 |

5|1, 2, 5, 1, 5 |

1, 2, 1, 1, 1 |

∴ L.C.M = 2 X 3 X 5 X 2 = 60

∴ Required number = 60 + 3 = 63

Q37. The L.C.M of (2^{2} X 3^{3} X 5 X 11^{2}), (2^{4} X 3 X 5^{2} X 11), and (2 X 3 X 5^{3} X 11^{3}) is

(a) 2^{2} X 3^{4} X 5^{2} X 11^{3}

(b) 2^{3} X 3^{3} X 5^{3} X 11^{2}

(c) 2^{4} X 3^{2} X 5^{2} X 11^{3}

(d) 2^{4} X 3^{3} X 5^{3} X 11^{3}

**Answer:** (d) 2^{4} X 3^{3} X 5^{3} X 11^{3}

2^{2} X 3^{3} X 5 X 11^{2} = 2 X 2 X 3 X 3 X 3 X 5 X 11 X 11

2^{4} X 3 X 5^{2} X 11 = 2 X 2 X 2 X 2 X 3 X 5 X 5 X 11

2 X 3 X 5^{3} X 11^{3} = 2 X 3 X 5 X 5 X 5 X 11 X 11 X 11

∴ L.C.M = 2 X 2 X 2 X 2 X 3 X 3 X 3 X 5 X 5 X 5 X 11 X 11 X 11

= 2^{4} X 3^{3} X 5^{3} X 11^{3}

Q38. The sum of two numbers is 50. Their difference is 1/5 of their sum. The L.C.M is

(a) 54

(b) 55

(c) 57

(d) 60

**Answer:** (d) 60

Let, the numbers be x and y

A/Q x + y = 50 --------- (1)

x - y = 50 X (1/5) = 10 ------ (2)

(1) + (2) ⇒ 2x = 60

⇒ x = 30

∴ y = 50 - 30 = 20

∴ No. are 30, 20

2|30, 20 |

5|15, 10 |

3, 2 |

∴ L.C.M = 2 X 5 X 3 X 2 = 60

Q39. The H.C.F and L.C.M of two numbers are 10 and 60 respectively. If the sum of the numbers is 50, then one of the number will be

(a) 5, 10

(b) 15, 20

(c) 20, 30

(d) 25, 35

**Answer:** (c) 20, 30

Let, the numbers be 10x and 10y, where x and y are co-primes

Now 10x X 10y = L.C.M X H.C.F

⇒ 10x X 10y = 10 X 60

⇒ xy = (10 X 60)/(10 X 10) = 6

Co-prime with product 6 are 2 and 3

∴ x = 2 y = 3

∴ Numbers are 10 X 2, 10 X3 = 20, 30

Q40. The H.C.F of three numbers is 10. If they are in the ratio 1 :2: 3, then the numbers are

(a) 5, 10, 15

(b) 10, 20, 30

(c) 15, 30, 45

(d) 20, 40, 80

**Answer:** (b) 10, 20, 30

Let the numbers be x, 2x, 3x

∴ H.C.F = x

Given that H.C.F = 10

⇒ x = 10

∴ The numbers are 10, 20, 30