17. | What will be the least number which when doubled will be exactly divisible by 12, 18, 21 and 30 ? | |||||||

L.C.M. of 12, 18, 21 30 2 | 12 - 18 - 21 - 30 ---------------------------- = 2 x 3 x 2 x 3 x 7 x 5 = 1260. 3 | 6 - 9 - 21 - 15 ---------------------------- Required number = (1260 ÷ 2) | 2 - 3 - 7 - 5 = 630. |

18. | The ratio of two numbers is 3 : 4 and their H.C.F. is 4. Their L.C.M. is: | |||||||

Let the numbers be 3 So, the numbers 12 and 16. L.C.M. of 12 and 16 = 48. |

19. | The smallest number which when diminished by 7, is divisible 12, 16, 18, 21 and 28 is: | |||||||

Required number = (L.C.M. of 12,16, 18, 21, 28) + 7
= 1008 + 7 = 1015 |

20. | 252 can be expressed as a product of primes as: | |||||||

Clearly, 252 = 2 x 2 x 3 x 3 x 7. |