Least common multiple

This reminds us of the fallacy in defining prime numbers.
Often it is said that a positive integer is prime if and only if it is divisible by one and itself.
I even heard Keith Devlin use that very definition. But of course according to that definition one is prime.
The correct definition is a positive integer is prime if and only if it has exactly two divisors.
 
JeffM said:
"One number is 18" That is not a variable at all, let alone an independent variable. What are "all the possibilities for the other number." That word "other" excludes 18. This is not math; this is what "other" means in English.
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The sum of 2 numbers is 36. One number is 18. What are the possibilities for the other number?
 
pka said:
This reminds us of the fallacy in defining prime numbers.
Often it is said that a positive integer is prime if and only if it is divisible by one and itself.
I even heard Keith Devlin use that very definition. But of course according to that definition one is prime.
The correct definition is a positive integer is prime if and only if it has exactly two divisors.
Click to expand...
one is divisible by one and itself, the problem is that "itself" means also one, so we are doing the same thing, so it is obvious that one can't be a prime number with this definition. But the example given by @lev888 means we are face to a typo :thumbup:

Thank you @lev888
 
JeffM said:
"One number is 18" That is not a variable at all, let alone an independent variable. What are "all the possibilities for the other number." That word "other" excludes 18. This is not math; this is what "other" means in English.
Click to expand...
English is my second language. But I don't see any ambiguities here caused by different meanings of words in different languages.
You earlier stated the problem as follows:
"The set of integers of which 18 is the least common multiple is indeed 1, 2 , 3, 6, 9, and 18.
One of the numbers is 18. What are the other numbers. It is a question about the meaning of the word "other."

The answer to this problem, of course, does not include 18. But this is not the original problem. There we are given 2 numbers, not a set 1, 2, 3, 6, 9, 18. We are told that one of the numbers is 18. We are asked about the other number - the other of the 2. Not of the 6 number set.
 
lev888 said:
The sum of 2 numbers is 36. One number is 18. What are the possibilities for the other number?
Click to expand...
Natural languages are filled with inconsistencies. If you construct a situation where it is impossible for there to be any alternative, then native speakers of English will interpret the combination of “two” and “other” to be two instances of the same thing, but in circumstances where there are alternatives, “two” and “other” will normally be interpreted as referring to different things.

In the original question, we were asked why the people who wrote the test interpreted it to designate the set 1, 2, 3, 6, and 9 as the correct answer rather than 1, 2, 3, 6, 9, and 18.

I explained why. It is how native speakers normally apply English when there are distinct alternatives. It has nothing to do with technical definitions in mathematics. Now the OP has decided that his mathematical analysis of English is correct. Natural languages are not mathematical or logical constructs.
 
JeffM said:
Natural languages are filled with inconsistencies. If you construct a situation where it is impossible for there to be any alternative, then native speakers of English will interpret the combination of “two” and “other” to be two instances of the same thing, but in circumstances where there are alternatives, “two” and “other” will normally be interpreted as referring to different things.

In the original question, we were asked why the people who wrote the test interpreted it to designate the set 1, 2, 3, 6, and 9 as the correct answer rather than 1, 2, 3, 6, 9, and 18.

I explained why. It is how native speakers normally apply English when there are distinct alternatives. It has nothing to do with technical definitions in mathematics. Now the OP has decided that his mathematical analysis of English is correct. Natural languages are not mathematical or logical constructs.
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I don't want to take more of your time on this, so feel free to not reply. I don't understand the first paragraph. In what way does this problem allow for no alternatives? Can you provide examples of “two” and “other” referring to the same/different things? Very confused here...

I agree that natural languages are not as formal as math notation. But I think that it is entirely possible to construct unambiguous problem statements in English. Otherwise, we simply would not have "word problem" as a type of math problems. And I am not even talking about legal matters...

You didn't address my example of the sum of 2 numbers. Or at least I couldn't identify it in your reply. So, what is the difference between the LCM problem and my example? In the sum, "other" clearly refers to the other number of the 2. What makes "other" in LCM not refer to the same thing?
 
I might have an answer... As this problem is given for GCSE pupils, the set we have to consider is [imath]\N[/imath], therefore the other number cannot be 18. If we consider the set [imath]\N\times\N[/imath], thus, 18 belongs to the set of solutions.

Anyway, I prefer to include 18...

What do you think?
 
darkyadoo said:
The LCM of two numbers is 18. One of the numbers is 18.

  1. Write down all the possibilities for the other number.
  2. Describe the set of numbers you have created.
The set I propose is [imath]\big\{1,2,3,6,9,18\big\}[/imath]

The correction doesn't include 18 ...? Should I include 18?

Thanks
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darkyadoo said:
My problem is not to find the solution, but to know why the Edexcel GCSE maths book excludes 18 of the set of solutions
Click to expand...

To me, you are obviously right, though I can understand their answer.

My justification is this: It is possible to find the LCM of two numbers that are the same; LCM(18, 18) = 18. So 18 is included.

Their justification would be that (in their context) "two numbers" implies "two different numbers", perhaps because they never ask for the LCM of 18 and 18, because it seems unnecessary. Possibly when they talk about adding fractions like 5/18 + 7/18, they would not ask for the LCD (the LCM of 18 and 18), because they already have a common denominator.

Mathematically, the answer has to include 18. Talking to students who aren't accustomed to mathematical thinking, which tends to be inclusive, it is understandable that 18 would be excluded. These are two different dialects.

It's also possible that they just didn't think of this possibility.

Can you show us the complete answer they give? I want to see how they describe the set (all divisors of 18? all proper divisors of 18?).
 
29071


That is their answer
 
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