It is extremely annoying to search back over pages to find what you quote in part so the whole context can be seen. This is especially true about a post that is over 4 months old; I cannot put myself back in the mental context of a post that old without reading everything that went before it. If you have questions about a post, please quote the entire thing and do so promptly.
What follows is the post that you asked the question about in red and my current comments in blue.
Suppose a, b, c, and d are positive whole numbers and that a * b = c
First, note that the realm of discourse is positive integers. When you start introducing fractions or zero, you are completely changing what is being discussed. This is why your jumping around willy-nilly gets you mired in "doubts." Your "doubts" are confusion arising from not sticking to a topic until you have learned it. If you ask a question about least common multiple and highest common factor, you are talking about positive whole numbers and dealing with number theory. Abstract algebra is completely off-topic and will only confuse the issue because abstract algebra is talking about things that may not even be numbers.
Second, a, b, c, and d are supposed to represent ANY positive whole numbers SUBJECT to whatever restrictions are specified. It is frequently helpful when dealing with general propositions to take a a specific example. I am going to restate what we are talking about.
[MATH]\text {Let } a,\ b, c, \text { and } d \text { be any positive whole numbers such that }\\
a * b = c \text { and } d < b.[/MATH]
Let's take as our example a = 3, b = 17, c = 51, and d = 5. Now any "doubts" that you may have should be capable of being expressed in an example THAT MEETS THE CONDITIONS SPECIFIED.
We are interested in what are the highest common factor and least common multiple of a and c.
The smallest multiple of c is c * 1 = c.
What I am saying here is that if I multiply c and any number OTHER THAN 1, the product (the multiple) will be larger than c. For example, 2 * 51 = 102 > 51. It's true of course that fifty-one times minus ten is less than fifty-one, BUT WE ARE NOT TALKING ABOUT NEGATIVE NUMBERS. It's also true that fifty-one times zero times is less than fifty-one, BUT WE ARE TALKING ABOUT POSITIVE NUMBERS, and zero is not a positive number. And finally it is true that fifty-one times a proper fraction is less than fifty-one, BUT WE TALKING ABOUT WHOLE NUMBERS.
In other words, because we are talking about positive whole numbers, out of the infinite number of multiples of 51, the smallest, the least, is 51 * 1 = 51. If you have doubts that the general proposition that c * 1 is the least multiple of c when we are talking about positive whole numbers, please give me the specific example that raises a doubt in your mind.
I have not yet claimed that c * 1 is the least COMMON multiple of a and c. We have not yet talked about the multiples of a. There are an infinite number of them. And the list of common multiples of a and c is also infinite.
If d < b then a * d < c.
What am I getting at here? I am going to consider the finite list of multiples of a that involve some positive whole number less than b (the list has only b - 1 items).
d < b \implies a * d < a * b because a is positive.
Therefore a * d < c because a * b = c.
Therefore a * d < c * 1 because c * 1 = c.
Therefore, a * d is a multiple of a, but it is less than c * 1, which is the least multiple of c. Therefore, a*d is not a COMMON multiple of a and c.
But a * b = c = c * 1 and
Therefore the least common multiple of a and c is c.
In my example, a = 3 and b = 17. Let's list the first 16 multiples of 3. 3, 6. 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48. Each of them is smaller than 51 so none of them can be a common multiple of 3 and 51 because the smallest multiple of 51 is 51 itself. So my remark in red above is justified by this example. If you can give me a counter-example, I shall willingly concede that your doubts have some basis.
What is the greatest whole number that divides a evenly. Obviously a itself. And by hypothesis a divides c evenly.
So the highest common factor is a.
Any questions or doubts on those two lines? We can then summarize with:
Therefore if a divides c evenly, a is the highest common factor and c is the lowest common multiple
We do not need to bother with prime factorization at all.
There are many other things we do not have to bother with such as the entire corpus of abstract algebra.
Just grasp what highest and lowest and common mean in context.
The helpers here will try to answer any questions you have about their own posts if you quote those posts fully and pose the questions promptly. It is unreasonable, however, to ask them to clarify what someone else has written. It is reasonable to ask a helper to resolve an apparent contradiction between the helper's post and a cited alternative authority IF YOU ARE SURE THAT THE ALTERNATIVE IS ON TOPIC.
