Student question to use common multiples ot discovery prime numbers, and the fundamental theory of arithmetic

I had a student say that you can figure out the second common multiple by doubling the LCM.

So far I cannot find a non example of this theory.

Do you know of any two numbers where this theory does not work???

Specific example of the student's claim:

Take two numbers. Say 3 and 4.
Find the first common multiple by listing all the multiples for each:

Common multiples of three are. 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... and
Common multiples of four are. 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Therefore, the first common multiple is 12, the second is 24, the third is 36, ...

Therefore, for this example the first common multiple is 12 and the second is 24 and 24 is twice 12.

So the students question is... Will the second common multiple (24 in this example) always be twice the first.

This is an excellent opportunity to use the Fundamental Theorem of Mathematics to help this student, and others, prove their conjecture.

One way to proceed:

We could find for each multiple in each of the sets of multiples (3 and 4) all the prime factors for each multiple.

My reasoning is that if I can find all the smallest factors that are only primes, then I may be able to see a pattern to generalize for every multiple.

So a list of all the smallest factors (primes) for the first few multiples of each multiple for: three and four.

0
3 - 3
6 - 2, 3
9 - 3, 3
12 - 2, 2, 3
15 - 3, 5
18 - 2, 3, 3,
21 - 3, 7
24 - 2, 2, 2, 3 ---> notice this is exactly the same as the prime factors of 12 * 2 or double. Try some more examples... Can make it infinite?
27 - 3, 3, 3
30 - 2, 3, 5
33 - 3, 11
36 - 2, 2, 3, 3 ---> notice this is exactly the same as the prime factors of 12 * 3 or triple.

0
4 - 2, 2
8 - 2, 2, 2
12 - 2, 2, 3 ---> notice....
16 - 2, 2, 2, 2
20 - 2, 2, 5
24 - 2, 2, 2, 3 ---> notice...
28 - 2, 2, 7
32 - 2, 2, 2, 2, 2
36 - 2, 2, 3, 3 ---> notice ...

NOW this is where the student might see the pattern and recognizes the answer to his or her question.

If they do or don't either way, it is still necessary to complete the next step, and illustrate the discovery of the fundamental theorem of arithmetic.

To get to that discovery, look at the prime factors for the first 25 whole numbers?

0 - 0
1 - 1
2 - 2
3 - 3
4 - 2, 2
5 - 5
6 - 2, 3
7 - 7
8 - 2, 2, 2
9 - 3, 3
10 - 2, 5
11 - 11
12 - 2, 2, 3
13 - 13
14 - 2, 7
15 - 3, 5
16 - 2, 2, 2, 2
17 - 17
18 - 2, 3, 3
19 - 19
20 - 2, 2, 5
21 - 3, 7
22 - 2, 11
23 - 23
24 - 2, 2, 2, 3
25 - 5, 5

Okay now, which of the numbers have exactly the same factors?
NONE?
WOW! Every whole number is a product of a unique set of prime numbers. In other words a group of primes has only one product. AND every whole number only has one set of prime numbers that can be used to make that number.

Now. If you are all still with me, put these two ideas together and think of
12 as the group of primes 2, 2, 3.
In order to get the next multiple, of that number, it would have to include another prime factor; and the smallest prime factor is 2. Oh, wow!
12 = 2 * 2 * 3 and multiply it by the smallest prime
2 * 2 * 3 * 2
and where was something like that seen before?
24 = 2 * 2 * 2 * 3

Therefore, if every whole number can be expressed as a unique group of primes, then any whole number (which would include any LCM for any group of numbers) would have as its next multiple one more factor and the smallest factor that it could be is two. SOoo, anyone that wants to find a counterexample that doesn't fit this students conjecture probably will have a long search with the odds being more unlikely of success than even winning the lottery.

YES! wasn't that fun?

Dr. Robert Sweetland's Notes ©