GCF-LCM Stuff: Find them: how, why, find smart and fast

What are common divisors and common multiples? Let’s learn this together with an example. Given two numbers 12 and 9, each has its divisors as:

Example: List the common divisors and common multiples of 12 and 9.

Divisors of 12: 1, 2, 3, 4, 6, 12.

Divisors of 9: 1, 3, 9.

Let integers N>0, M>0. Recall quickly that a divisor of N is a positive integer (counting number) d<N such that d divides into N evenly (that is, remainder is zero). Given N and M, a common divisor is one divisor of N and of M both.

In the example above, the common divisors of 12 and 9 are 1 and 3, while 1 is the least, and 3 is the greatest – usually written as GCD (Greatest Common Divisor) or GCF (Greatest Common Factor).

Now let’s turn our attention to the common multiples.

Multiples of 12: 12, 24, 36, 48, 60, 72, ..

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, ..

Recall quickly a multiple of N can be obtained by N multiplied by any positive integer. Given N and M, a common multiple of N and M, as said in its name, is the common multiple of N and of M both.

In the example above, the common multiples of 12 and 9 are 36, 72, etc. 36 is the least one, usually referred as LCM (Least Common Multiple).

This discussion also gives us one way of finding GCF and LCM: just list common factors /divisors of each and common multiples of each, then circle the greatest of common factors and the least of common multiples. This way, of course, stick strictly to the definition.

Is it a better way to find the GCF and LCM of two numbers?

Yes, indeed. We know that 9 = 3 x 3. Let’s check whether 12 can be divided by 3 (Yes) and by 9 (No). We can conclude that GCF of the two numbers is 3.

How about LCM? Let us divide 12 by 3 (which is the GCF) to get 4, and multiplies by the other number 9, we get 36 – which is the LCM. Or we can divide 9 by the GCF 3, and multiplies by the other 12, again we get 12 the LCM.

Wow! This seems much easier. Why this approach works?

We’d better do some investigation to find out – but before doing that, let’s verify by another example.

Find the GCF and LCM of numbers 108 and 40.

Ohh, GCF is 4 and LCM is 1080. Have you guessed it correct?

One hint for why the approach works is, in the above two examples, if you multiply the GCF and LCM, you will find the product to equal exactly the product of the two given numbers, as:

9 x 12 = 3 [GCF] x 36 [LCM], and 40 x 108 = 4 [GCF] x LCM [LCM].

Based on this there are also games (GCF-LCM Web), would you like to have a try?

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