Common Muliples

HallsofIvy said:
Yes, since 8 is the smallest number in the both the "4" and "8" rows 8 is the smallest common multiple of 4 and 8.
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But my confusion is that in the table of data each set of numbers have two rows, i.e. row 1 shows 2,4,6,8.... then row 2 shows 3,6,9,12...

Now looking at lowest common multiple of 2 and 3, I can use rows 1 and 2 for comparisons and see that 6 occurs in both rows as a lowest common multiple of each number 2 and 3.

Then I can look at LCM of 4 and 6, and if I look in rows 3 and 4 I can see my lowest common multiples are 12 in each row.

What I'm saying and showing is that the last row, row 5 which has data; 8,16,24,32... does not have another row after it for comparison data.

So although I know that the LCM of 4 and 8 is 8, I don't think the table of data shows me that solution?
 
I would not put the multiples into a table, because columns are irrelevant to the task. I would just write lists:

2, 4, 6, 8
3, 6, 9, 12
4, 8, 12, 16
6, 12, 18, 24
8, 16, 24, 30

You only need to look in the rows corresponding to the numbers of interest; I'm not sure why you think you need to look at the next row. To find LCM(4, 8), you just look in the rows listing multiples of 4 and 8, looking for the smallest number that is in both:

4, 8, 12, 16​
8, 16, 24, 30​
 
So at this moment in time I'm not sure how to read the data correctly. I only put the data into a table to keep it presentable and readable. As long as we understand everything I'm doing is a learning curve for me and not really revision as such. I know I've covered these topics I post in years gone by but I want to try and learn the understanding better this time round.

Looking at my information the data rows and columns (not in a table) advise to look for comparison data, like in your row (2) there is a number 12, and in row three there is a number 12 again, so I'm reading the data from top to bottom for comparisons. At that point there I'm saying that the LCM of 3 and 4 is 12. I think you are therefore saying that I can look at my LCM's say 4 and 8 and look at any of the rows where 4 and 8 are in, and then as your example shows reading from the left hand side I can say that the LCM of 4 and 8 is 8.

Am I getting this understanding correct?
 
Probability said:
Looking at my information the data rows and columns (not in a table) advise to look for comparison data, like in your row (2) there is a number 12, and in row three there is a number 12 again, so I'm reading the data from top to bottom for comparisons. At that point there I'm saying that the LCM of 3 and 4 is 12. I think you are therefore saying that I can look at my LCM's say 4 and 8 and look at any of the rows where 4 and 8 are in, and then as your example shows reading from the left hand side I can say that the LCM of 4 and 8 is 8.

Am I getting this understanding correct?
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No. What did I say? "Just look in the rows listing multiples of 4 and 8." That means the rows that begin with 4 and 8, not just any rows containing 4 and 8.

If you think about what LCM means, this should be clear. The LCM of 4 and 8 is the Least Common Multiple: that is, the Smallest number that is a Multiple of Both 4 and 8. So we list the numbers that are multiples of 4 (4, 8, 12, 16, ...), and the numbers that are multiples of 8 (8, 16, 24, 32, ...), and find the smallest number that is in both lists, namely 8 itself.

I don't need to look at lists of multiples of anything else; these two rows in your table are all I need. I don't "look top to bottom"; I only look left to right. And if LCM(4, 8) is all I want, I won't write any other rows. (In reality, I wouldn't write any rows, because I have other methods; but this method is the best way to understand the meaning of LCM.)
 
Continuing on with this subject just a little while longer until I feel I have properly understood it.

I have a question.

A chocolate manufacturer wants to produce chocolates that can be shared evenly among people in any group of four or fewer people.

The question asks;

What is the smallest number of chocolates that can be in the box?

I'm not so sure I agree with the answer provided, which is 12.

I have four people. If the box of chocolates contained 4 chocolates then each person could have 1 chocolate each, therefore the chocolates have been shared out evenly at 1 chocolate each. If each person had 2 chocolates each then the box could contain 8, and if the box contained 12 then each person could have 3 chocolates each. As we are asked to find the smallest number, then a box of chocolates containing 12 is not the smallest number that can be shared to me.

I'm not sure why the questions says four or fewer people but then does not start the math problem with 1 person?

Is it because the minimum people that can share the chocolate in the group of 4 is 2?

So in keeping with common multiples;

I have four people;

The common multiples are;

2, 4, 6, 8, 10, 12
3, 6, 9, 12
4, 8, 12

LCM = 12

Smallest number of chocolates that can be in the box = 12.

Question is why?
 
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If you have 12
then it can be shared by 4 people evenly
then it can be shared by 3 people evenly
then it can be shared by 2 people evenly
then it can be shared by 1 person evenly.


If you replaced 12 with 1,2,3,4,5,6,7,8,9,10 or 11 you can NOT say that the chocolate can be shared by 4, 3, 2 AND 1 person.

12 is the smallest number that works.

You need to compute the lcm(1,2,3,4) = lcm(2,3,4) = lcm (3,4) since 2 goes into everything that 4 goes into.
 
I'm not really sure I understand Jomo to be honest. the way I think I understand it now is that I have a maximum of four people and a minimum of two people if I want to share the chocolates. 1 person can't come into the math problem as I see it because 1 person cannot share the chocolate.

I can either share the chocolate between 2, 3 or 4 people in the group. Remember it says four people or fewer.

The math is looking at it mathematically, but the minimum amount of chocolate in a box for up to four people could be 1 chocolate each?
 
I disagree that 1 person can't share the chocolate. The good part is that doesn't have to matter.

If you tried what I suggested you would have seen that if you have 2 pieces of chocolate then you can not give 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

if you have 3 pieces of chocolate then you can not give 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

if you have 4 pieces of chocolate then you can not give 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

if you have 5 pieces of chocolate then you can not give 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

...

if you have 11 pieces of chocolate then you can not give 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

if you have 12 pieces of chocolate then you CAN GIVE 4 people, 3 people AND 2 people a whole number of pieces of chocolate.

So the answer is 12.

You said in your last post that you did not understand my post. Please tell us exactly which part you do not understand in that post, this post or any other post.
 
I need to understand your question. It would be helpful if you would clearly state what you got from somewhere else, and what you are saying yourself.

I think what is in red is the problem given to you, what is in green is the solution given to you (from where?), and the rest is your own:

Probability said:
I have a question.

A chocolate manufacturer wants to produce chocolates that can be shared evenly among people in any group of four or fewer people.

The question asks;

What is the smallest number of chocolates that can be in the box?

I'm not so sure I agree with the answer provided, which is 12.

I have four people. If the box of chocolates contained 4 chocolates then each person could have 1 chocolate each, therefore the chocolates have been shared out evenly at 1 chocolate each. If each person had 2 chocolates each then the box could contain 8, and if the box contained 12 then each person could have 3 chocolates each. As we are asked to find the smallest number, then a box of chocolates containing 12 is not the smallest number that can be shared to me.

I'm not sure why the questions says four or fewer people but then does not start the math problem with 1 person?

Is it because the minimum people that can share the chocolate in the group of 4 is 2?

So in keeping with common multiples;

I have four people;

The common multiples are;

2, 4, 6, 8, 10, 12
3, 6, 9, 12
4, 8, 12

LCM = 12

Smallest number of chocolates that can be in the box = 12.

Question is why?
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Am I right about that? I'm guessing this because the only place it suggests a minimum number of people as 2 in the green part, and if that were your own thinking, you wouldn't be questioning it.

We don't normally say we "share" something with only ourselves; that's the only reason they wouldn't mention 1 person! But that has no effect on the answer, because any number is divisible by1. Any number less than 12 would not be divisible by 1, 2, 3, and 4, while 12 is, so clearly that is the smallest such number.

What other answer do you think would be correct if you allowed for one person? How does that change your answer?

Probability said:
The math is looking at it mathematically, but the minimum amount of chocolate in a box for up to four people could be 1 chocolate each?
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I wonder if you are misinterpreting the whole question. "1 chocolate each" is not an "amount of chocolate in a box". We are looking for a number in the box that can be divided by 1, 2, 3, or 4. In each case, each person will get a different number of the 12 chocolates (namely 12, 6, 4, or 3 respectively), but all that matters is that this is a whole number in each case.

The important thing is not whether you can share with one person; it's whether you can share fractions of a chocolate.
 
Suppose you were going to have a party and you have a box of chocolate.

If three people come (including you), then three people share equally the chocolate.

If no one comes, except you, then you get to eat the chocolate. That is you share the chocolate with yourself. I can see how one may say that this is not sharing but I think it is.

The rule was that who ever comes to the party shares equally the box of chocolate. It is no ones fault if you are the only one who attends the party! I guess you can't even call it a party!
 
Let's take it one step at a time gents.

The question;

A chocolate manufacturer wants to produce chocolates that can be shared evenly among people in any group of four or fewer people.

What is the smallest number of chocolates that can be in the box?

I said;

I'm not so sure I agree with the answer provided, which is 12.

For me I think the problem here is trying to understand it mathematically, but what then confuses me is that I know 4 people can share evenly a box of chocolates with as few as 8 chocolates in.
 
Yes, 4 people can share a box with 8 chocolate in it.

But can 3 people do that?

Can 2 people do that?

If you say yes to ALL the above then you found a smaller common multiple than 12. But is 8 the least common multiple OR is it a common multiple at all?
 
Probability said:
The question;

A chocolate manufacturer wants to produce chocolates that can be shared evenly among people in any group of four or fewer people.

What is the smallest number of chocolates that can be in the box?

I said;

I'm not so sure I agree with the answer provided, which is 12.

For me I think the problem here is trying to understand it mathematically, but what then confuses me is that I know 4 people can share evenly a box of chocolates with as few as 8 chocolates in.
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You are misunderstanding the question.

It is looking for ONE box size that can be shared by ANY group of 1, 2, 3, or 4. Yes, a box of 8 could be shared by 1, 2, or 4, but not by 3; in fact, a box of 4 could be shared by 1, 2, or 4. But If it can't be shared also by 3, then it is not an answer to the question.
 
Jomo said:
Yes, 4 people can share a box with 8 chocolate in it.

But can 3 people do that?

Can 2 people do that?

If you say yes to ALL the above then you found a smaller common multiple than 12. But is 8 the least common multiple OR is it a common multiple at all?
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No your correct. I think this is a division problem?
 
Probability said:
For me I think the problem here is trying to understand it mathematically, but what then confuses me is that I know 4 people can share evenly a box of chocolates with as few as 8 chocolates in.
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For the record, 4 people can share evenly a box of chocolates with as few as 8 chocolates in. Why??
 
Dr.Peterson said:
You are misunderstanding the question.

It is looking for ONE box size that can be shared by ANY group of 1, 2, 3, or 4. Yes, a box of 8 could be shared by 1, 2, or 4, but not by 3; in fact, a box of 4 could be shared by 1, 2, or 4. But If it can't be shared also by 3, then it is not an answer to the question.
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Your completely correct I am misunderstanding the question, and at the moment I just can't get it clear in my head how to mathematically work it out. I can multiply out the numbers 2,3 and 4 until I gain 12 and say that the LCM = 12, but 1 x... will never equal 12 following the multiples rules.
 
This is how I'm seeing it to now...

Using multiplication from 1 to 4 to reflect the number of people in the problem.

1 x 1 = 1, 2 x 1 = 2, ..... 12 x 1 = 12

Your first line of data is;

1,2,3,4,5,6,7,8,9,10,11,12

Then you do the two times table and get;

2,4,6,8,10,12

You do the three times table and the four times table and the results are as follows;

1 = 1,2,3,4,5,6,7,8,9,10,11,12
2 = 2,4,6,8,10,12
3 = 3,6,9,12
4 = 4,8,12

Looking at the data now for the smallest amount of chocolate I see;

In this mathematical problem 1 cannot to me be shared by up to four people. the number 1 does not appear in any other list of data, i.e. rows 2,3 and 4.

Looking at all four rows of data, the only common number in there to all four rows is 12.

So from a mathematical point of view the LCM = 12.

To me looking at the data above, 1 person alone without other people cannot share the chocolate, there is nobody to share it with.
 
First, you need to separate the question of whether the word "share" is appropriate here, from what the LCM is. You keep seeming to pull them apart, but then they intermingle again. In this post, I will talk only about the language issue.

On the meaning of "share", you said:
Probability said:
To me looking at the data above, 1 person alone without other people cannot share the chocolate, there is nobody to share it with.
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The problem was:
Probability said:
A chocolate manufacturer wants to produce chocolates that can be shared evenly among people in any group of four or fewer people.
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The idea of sharing here is not "I want to share this candy with my friends", but "we want any group of four or fewer to be able to share a box." They are all sharing the box, dividing it evenly among them. The number each will get is the total number in the box, divided by the number of people in the group.

So they want the number in the box to be a multiple of 1 (which is true of any integer, so it's trivial), or of 2 ( so two people can each get the same whole number of them), or of 3, or of 4.

Do you understand that 12 is the answer to this question? (Pay no attention to words like LCM.) Do you see that 1 person will get all 12, 2 people will each get 6, 3 will each get 4, and 4 will each get 3? Do you see that any smaller number would not work?

We'll get to the abstract question of LCM (if you still need it) after you are satisfied with the answer to the word problem concretely.
 
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