LCM stands for Least Common Multiple.
And it is defined as the product of the two numbers divided by the GCD (greatest common divisor)
In our case, the product of 4 and 6 is 24, , and the greatest common divisor of 4 and 6 is "2". Therefore, the LCM of 4 and 6 is 24/2 = 12
Let me also use the Venn diagram that your teacher provided:
In the diagram we enter the factors that correspond to both numbers (4 and 6), and in the intersection of the two sets (intersection of the circle) we type a "2" which is the ONLY factor 4 and 6 have in common (the greatest common divisor of the two given numbers) So complete a diagram as follows:
We typed a 2in the area common to both numbers. Then your LCM is the product of 2 times 2 times 3 = 12
Notice the blue set (circle) contains the two factors for 4 (2 * 2) and the orange circle contains the two factor for 6 (2 * 3)
We set in the intersection of the two circles the factor that is common to both.
Do you want me to complete the second question with a Venn diagram as well? Perfect.
The second question is about the LCM of the numbers 12 and 8
Then we create a Venn diagram like the following, considering that the factor in common between 12 and 8 is 4, because 12 = 4 * 3 and 8 = 4 * 2
Again here, the factors 3 and 4 (that give 12) are typed in the blue circle. and the factors that form 8 (4 * 2) are typed inside the orange circle.
The factor that both share is in the middle "4". Therefore, now to find the LCM you simply multiply the three numbers shown in the Venn diagtam: 3 * 4 * 2 = 24
Then 24 is your LCM.