What is the LCM of 45 60 and 75?

Answer 1

To get the LCM of three numbers, we first need to find the LCM of 2 sets of 2 of these numbers.

LCM(60,75) = (60 x 75) / GCF(60,75) = 4500 / GCF(60,75)

60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5

75 = 3 x 25 = 3 x 5 x 5

GCF(60,75) = 3 x 5 = 15

LCM(60,75) = 4500 / 15 = 300

Then find the LCM of 45 and 60

LCM(45,60) = (45 x 60) / GCF(45,60) = 2700 / GCF(45,60)

45 = 3 x 15 = 3 x 3 x 5

60 = 2 x 30 = 2 x 2 x 15 = 2 x 2 x 3 x 5

LCM(45,60) = 2700 / 15 = 180

Then find the LCM of your two LCMs (180 and 300)

LCM(180,300) = (180 x 300) / GCF(180,300)

180 = 3 x 60 = 3 x 3 x 20 = 3 x 3 x 2 x 10 = 3 x 3 x 2 x 2 x 5

300 = 3 x 100 = 3 x 2 x 50 = 3 x 2 x 2 x 25 = 3 x 2 x 2 x 5 x 5

The common factors are 3, 2, 2, and 5, so:

GCF(180,300) = 3 x 2 x 2 x 5 = 60

Plug this back in.

LCM(180,300) = (180 x 300) / 60 = 54000 / 60 = 900

Finally, check to make sure 45, 60, and 75 go into this number.

900 / 45 = 20

900 / 60 = 15

900 / 75 = 12

To be sure that this is the true LCM, there should be no GCF (other than one) of the three numbers that you just got (12, 15, and 20)

12 = 2 x 6 = 2 x 2 x 3

15 = 3 x 5

20 = 2 x 10 = 2 x 2 x 5

There is no factor that is common to all three, so this is the true LCM.

This confirms that the answer is 900.

Answer 2

LCM is 900, which is the multiple of the highest power of prime factors in the given numbers (22 x 32 x 52).