Lowest Common Multiple (lcm)

Notes

The lowest common multiple (LCM) is defined by the smallest integer multiple that is shared by two or more numbers, e.g., the LCM of 5 and 8 is 40.

COMMON MISTAKE: In the above, the LCM was calculated by taking the product of the given numbers. However, this method does NOT apply to all contexts, e.g., the LCM of 6 and 8 is NOT 48 it is 24.

v SOLUTION: The ONLY scenario where you can be 100% sure that the LCM is a product of the given numbers is where the given numbers are all prime numbers. Otherwise, make sure to proceed from Step 1) outlined below

The most efficient method for calculating the LCM would involve drawing tree diagrams and a Venn diagram. Follow the steps outlined below:

Step 1) Draw the tree diagram (of the given numbers) as a product of its prime factors, e.g., 32 as a product of its prime factors is written as 2 * 2 * 2 * 2 * 2 = 2⁵

NOTE: A prime number is a number that produces an integer output only when it is divided by itself and 1. Therefore, when a given number is written as a product of its prime factors, by definition, they cannot be simplified to be written as a product of more factors.

Step 2) Place each factor in the correct part of the Venn diagram with common factors (between the given numbers) being placed in the intersection of the circles drawn

Step 3) Multiply the numbers from all sections of the circles (i.e., the common prime factors between the given numbers and their unique prime factors) to find the LCM

Question 1) Find the Lowest Common Multiple of 33 and 48

Question 2) Compute the Lowest Common Multiple of 39 and 85

Question 3) Work out the Lowest Common Multiple of 17 and 19

Question 4) Find the Lowest Common Multiple of 155 and 252

Question 5) Calculate the Lowest Common Multiple of y and 8y

Answer 1)

33 = 11 * 3

48 = 2 * 24 = 2 * 2 * 12 = 2 * 2 * 2 * 6 = 2 * 2 * 2 * 2 * 3 = 3 * 2⁴

So, LCM = 2⁴ * 3 * 11 = 528

Answer 2)

39 = 3 * 13

85 = 5 * 17

So, LCM = 3 * 13 * 5 * 17 = 3,315

NOTE: There are NO prime factors that lie in the intersection of 39 and 85 in the Venn diagram

Answer 3)

17 = 17

19 = 19

So, LCM = 17 * 19 = 323

NOTE: Since 17 and 19 are both prime numbers, the LCM is simply their product

Answer 4)

252 = 2 * 126 = 2 * 2 * 63 = 2 * 2 * 21 * 3 = 2 * 2 * 7 * 3 * 3 = 3² * 2² * 7

155 = 5 * 31

So, LCM = 3² * 2² * 7 * 5 * 31 = 39,060

NOTE: There are NO prime factors that lie in the intersection of 39 and 85 in the Venn diagram

Answer 5)

LCM of y and 8y = 8y