20230204, 17:53  #1 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2×229 Posts 
Definition of “common multiple”
I am thinking of doing an Open University (OU) mathematics degree.
https://www.open.ac.uk/courses/maths...athematicsq31 This is parttime, distance learning. I would need to start on the course “Essential Mathematics 1”, which is pretty basic. https://www.open.ac.uk/courses/modules/mst124 The OU fees include all the books, but I decided to buy the course books from eBay, to start reading them now. I am going through book A but are confused by this statement “A common multiple of two or more integers is a number that is a multiple of all of them. For example the common multiples of 4 & 6 are …, 36, 24, 12, 0, 12, 24, 36, … Common multiple is not a term that I have come across before, although I have come across least common multiple (lcm). I decided to Google the term common multiple and found this webpage https://www.splashlearn.com/mathvoc...ommonmultiple There they list the common multiples of 6 & 7, and say 42 & 84 are the common multiples. Is this an ambiguous term, or are one of the sources giving blatantly wrong information? 
20230204, 19:09  #2 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2377_{10} Posts 
Where is the conflict between the 2?
Can you be specific of what number is included in one but not the other. 
20230204, 19:15  #3  
Apr 2020
3FF_{16} Posts 
Quote:
In practice we often work only with positive integers, so it's perfectly fine to say that the common multiples of 6 and 7 are 42, 84, 126, 168, ... ignoring 0 and the negative common multiples. 

20230204, 19:36  #4 
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2×229 Posts 
Ignoring 0 and negative integers, the OU source gives 12. But 6 x 4 = 24, not 12. So how do the OU say 12 should be in the list?
I appreciate that https://www.splashlearn.com/mathvoc...ommonmultiple is just not listing larger numbers. That's fair enough  the full list would be infinite in length. But the lowest value they give (42) is obtained by multiplying 6 by 7. But the OU gets a smaller number (12) than multiplying 6 and 4. Last fiddled with by drkirkby on 20230204 at 19:41 
20230204, 19:48  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2377_{10} Posts 
I think you are correct. If you include 12 for 6 and 4, then there is no logical way to exclude 8 which is not included. I would let it down as a typo since there is no logical/algorithmic way of generating:
0, 12, 24, 36, without including 8. Nice catch. My old eyes/brain failed me yet once more. I would define commonmultiples of integers a and b as all the multiples of a*b where the multiplier can be any integer. But I am usually wrong about everything. So take that with a grain of sugar (or is it sand). Last fiddled with by a1call on 20230204 at 19:53 
20230204, 19:56  #6  
Apr 2020
1023_{10} Posts 
Quote:
lcm(m,n) = mn/gcd(m,n) where gcd = greatest common divisor (sometimes known as highest common factor). Quote:
Quote:
Last fiddled with by charybdis on 20230204 at 19:59 

20230204, 20:00  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2,377 Posts 
I stand corrected thank you charybdis.
ETA: Hay (or it Hey). two things, I was correct about the grain of sugar too. ETA II: This helped me understand the algorithm: https://multiply.info/CommonMultiple...f4and6.html Last fiddled with by a1call on 20230204 at 20:16 
20230204, 20:23  #8  
"David Kirkby"
Jan 2021
Althorne, Essex, UK
111001010_{2} Posts 
Quote:
I note that maxima gives lcm(6,4) as 12 and not 24. 

20230204, 20:33  #9  
"Rashid Naimi"
Oct 2015
Remote to Here/There
100101001001_{2} Posts 
Quote:
Factoring m and n could be computationally expensive. It's ETA: You wouldn't need to factor the GCD, unless you wanted to include the likes of 8 in relation to 4 and 6 (which with this definition you don't). Last fiddled with by a1call on 20230204 at 20:50 

20230204, 20:52  #10  
"David Kirkby"
Jan 2021
Althorne, Essex, UK
2·229 Posts 
Quote:
https://www.open.ac.uk/courses/modules/mst124 I think this should have be explained better. But thank you both for your help. I have gone from being totally confused to feeling I understand it within a couple of hours! Last fiddled with by drkirkby on 20230204 at 20:53 

20230204, 21:52  #11 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2,377 Posts 
My pleasure drkirkby, anytime. I guess charybdis deserves some credit too.

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