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P Pankaj Sanodiya
(a) LCM = 2 2 5 = 20 (b) LCM = 2  3  3 = 18 (c) LCM = 2 2 2 2 3 = 48 (d) LCM = 3 3 5 = 45 Yes, it can be observed that in each case, the LCM of the given numbers is the larger number. When one number is a factor of the other number, their LCM will be the larger number.

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P Pankaj Sanodiya
(a) LCM = 2 2 3 3 = 36 (b) LCM = 2 2 3 5 = 60 (c) LCM = 2  3 5 = 30 (d) LCM = 2 3 5 = 30 Yes, it can be observed that in each case, the LCM of the given numbers is the product of these numbers. When two numbers are co-prime, their LCM is the product of those numbers. Also, in each case, LCM is a multiple of 3.

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P Pankaj Sanodiya
LCM of 18, 24 and 32 LCM = 2 2 2 2 2 3 3 We have to find the smallest 4-digit multiple of 288. It can be observed that 288 3 = 864 and 288 4 = 1152. Therefore, the smallest 4-digit number which is divisible by 18, 24, and 32 is 1152.

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P Pankaj Sanodiya
LCM of 6, 15, 18 LCM = 2 2 3 5 = 90 Required number = 90 + 5 = 95

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P Pankaj Sanodiya
Maximum capacity of the required tanker =HCF of 403,434,465 403=13 31 434=2 7 31 465=3 5 31 HCF=31  Hence the maximum capacity of a container that can measure the diesel of the three containers the exact number of times is 31.

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P Pankaj Sanodiya
The time period after which these lights will change = LCM of 48, 72 and 108 LCM=2 2  2 2 3 3 3 = 432 They will change together after every 432 seconds i.e., 7 min 12 seconds. Hence, they will change simultaneously at 7:07:12 am.

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P Pankaj Sanodiya
CM of 8, 10 and 12 LCM = 2   2   2   3 5 = 120 We have to find the greatest 3-digit multiple of 120. It can be seen that 120 8 = 960 and 120 9 = 1080. Hence, the greatest 3-digit number exactly divisible by 8, 10, and 12 is 960.

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P Pankaj Sanodiya
Smallest number = LCM of 6, 8, 12 We have to find the smallest 3-digit multiple of 24. It can be seen that 24   4 = 96 and 24   5 = 120. Hence, the smallest 3-digit number which is exactly divisible by 6, 8, and 12 is 120.

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P Pankaj Sanodiya
  Length =825cm=3 5 5 11  Breadth =675cm=3 3  3 5 5  Height =450cm=2 3 3 5 5  Longest tape = HCF of 825,675, and 450=  3 5 5                       =75cm  Therefore, the longest tape is 75cm.   

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P Pankaj Sanodiya
  Step measure of 1 st  Boy =63cm  Step measure of 2 nd  Boy =70cm  Step measure of 3 rd  Boy =77cm  LCM of 63,70,77 :

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P Pankaj Sanodiya
Weight of the two bags = 75 kg and 69 kg Maximum weight : HCF (75, 69) 75 = 3 5 5 69 = 3 23 HCF = 3 Hence maximum value of weight which can measure the weight of the fertiliser exact number of times is 3.
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