What is the difference between these two sets of answers for these two questions?

[lookagain]

my challenge question


1) What two positive integers are there such that their sum equals their product?


2) What is the positive integer such that when it is added to itself equals when it
is multiplied by itself?

 

[lookagain]

Does the "title" belong with the problem statement, <------ Yes.
and "my challenge question" should be the "title"? <------ That phrase should be part of the title.

Edit: I agree with your answers.

 

[daon2]

Questions like this can be better phrased to avoid confusion. I assume, possibly incorrectly, this is in reference in another recent thread where distinctness was not specified.

If the person asking the question requires distinctness #1 would have no solution. Whether or not "two numbers" and "a set of two numbers" are equivalent items (I feel) is up to interpretation.

An unambiguous way to ask this is: Find positive integers x and y such that xy=x+y. <--- here the answer is best given as a set of ordered pairs (x,y)

The second question is clear. Find a positive integer x such that x*x=x+x. <- here, distinctness is obvious

 

[lookagain]

Questions like this can be better phrased to avoid confusion. I assume, possibly incorrectly,
this is in reference in another recent thread
where distinctness was not specified. If distinctness is not specified, do not assume it.

If the person asking the question requires distinctness #1 would have no solution.
Whether or not "two numbers"
and "a set of two numbers" are equivalent items (I feel) is up to interpretation.

An unambiguous way to ask this is: Find positive integers x and y such
that xy=x+y. <--- here the answer is best given as a set of ordered pairs (x,y)

The second question is clear. Find a positive integer x such that
x*x=x+x. <- here, distinctness is obvious

No distinctness is stated in #1, and because addition and multiplication of the
numbers is commutative, no ambiguity can exist.
There is no "which one is which?" The statement of the question is sufficient.

The same would be true if the dimensions (without mention of a distinct width
or length) were asked of a rectangle of. say, X square units.

Stating that the dimensions of the rectangle are, say, Y units by Z units
is sufficient.


Here is an example of a question where order would matter:

"What are two positive integers in which one of them raised to the other equals 9?"

An answer of "2 and 3" is ambiguous, because \(\displaystyle x^y \ne y^x \ \ in \ \ general.\)

We would need to state "The base is 3 and the exponent is 2."


Or, given that x and y are integers greater than 1, determine x and y, such
that \(\displaystyle x^y \ = \ 9.\)

Then an answer could be (x, y) = (3, 2).