Is it true that to finding the lcm you have to divide the product but he gcf? if not give in example why not, and if it’s true, why does it work?

To find the LCM, you CAN divide the product of the two numbers by the GCF.

This is not a very well-known fact, but it is useful at times, especially when we are asked to calculate both the LCM and GCF.

For example, the GCF of 18 and 48 is 6, by inspection or by factors.

Therefore the LCM of 18 and 48 is 18*48/6=864/6=144.

It works because by definition the GCF is the products of all common factors between the two given numbers.  Dividing the product by the GCF will therefore remove the duplication of the common factors, leaving a number which contains the factors of each of the two numbers, thus a lowest common multiple.

Note that this method is not the only way to find LCM's.