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On a 4th grade test the question was asked:

17 is a multiple of only two numbers, 1 and 17. Tell why this statement is true.

If it is true then every number must be a multiple of 1 since 1 is a factor of every number. Right?

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closed as off-topic by Benjamin Dickman, Mark Fantini, JPBurke, user173, Markus Klein Oct 29 '14 at 11:30

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Welcome to the site, Donna. I hope you find my answer helpful to your situation. Please let us know if there's another aspect of the situation you'd like to think about with our help.

Test question: "17 is a multiple of only two numbers, 1 and 17. Tell why this statement is true."

I think they are asking the student to show that 17 is not a multiple of any other numbers. To do so, one can show that dividing by 2,3, ... always leaves a remainder.

I think you are asking whether it is right to conclude that "every number must be a multiple of 1 since 1 is a factor of every number".

Yes, every whole number is a multiple of 1. We say that b is a multiple of a when a*n = b (where n is a whole number). Since 1*b = b, for any number b, all numbers are multiples of 1.

It sounds like you also want to double-check your understanding of the two words, 'factor' and 'multiple'. If b is a multiple of a, then a is a factor of b. The two terms describe the same situation from different perspectives.

Is this helpful to you?

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Yes, every number and every thing is a multiple of one. 2 is. 5 is. 0.1 is. Potato salad is. Seriously, one times potato salad is still potato salad. Multiplying by one does nothing and you can do nothing to anything. And this has almost nothing to do with answering the test question. It just complicates the way it has to be asked. Answering the test question goes like this:

Because 17 is a PRIME number.

The word in the test question to obsess on here isn't multiple, or factor, it's ONLY.

BTW, the test question, as quoted, is actually false. It needs to be corrected to read:

17 is a multiple of only two whole numbers, 1 and 17. Tell why this statement is true.

Because there are an infinite number of numbers that can be multiplied together to give you 17: 1.7 x 10, sqrt(17) x sqrt(17), (17/2) x 2, etc. But there are only two whole numbers. That is why 17 is called a prime number. Any number that has only two whole number multiples is a prime number.

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    $\begingroup$ To make a thing a multiple of one, you need to define some kind of multiplication. If you define multiplication by one to keep everything intact, then everything is a multiple of one. But this answer seems to digress from the question, which seems to be about only natural numbers where "multiple" implicitly means "integer multiple". $\endgroup$ – Joonas Ilmavirta Oct 28 '14 at 11:02
  • $\begingroup$ Yes I am digressing. Because the posted question and the test question are actually dealing with different issues. I've tried to resolve both. $\endgroup$ – candied_orange Oct 28 '14 at 19:23
  • $\begingroup$ +1 for potato salad, my favorite math teacher actually used "cow" in these situations, I thought that was cool. Your edit "whole" is a good suggestion, but the question is a quote, and that's what we have to deal with. We can edit the OP's question, but not the quoted section. In my opinion. $\endgroup$ – JoeTaxpayer Oct 29 '14 at 13:11
  • $\begingroup$ Thank you, multiplying cow by 1 certainly works as well. Watch and be amazed as I do it to your checking account! Whooo! See how each number is still the same? Wish that worked with 2. In a universe that considers a fraction a number the test question is just plain false. I'd give any student that called me on that full marks if not extra credit. Nothing implies integers here. Expecting that to be understood is just making the student play, "guess what I'm thinking". $\endgroup$ – candied_orange Oct 29 '14 at 14:10
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This may or may not be a "4th grade problem" (but I think it is), but the natural numbers (the counting numbers or ordinal numbers) are defined by $1$. $2$ is "defined" as $1 + 1$, $3$ is "defined" by $1 + 1 + 1$...$17$ is "defined" by $1 + 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1$.

In answer to your actual question: If $17$ is a multiple of only two numbers, $1$ and $17$, is it true that all numbers are multples of $1$, then I would answer no!

That information alone is not enough to infer that all number are multiples of $1$. Frankly your question is quite circular: "If it is true then every number must be a multiple of 1 since 1 is a factor of every number. Right?"

If it is true that every number is a multiple of $1$, then yes, it's virtually trivial to prove that every number is a factor of $1$.

Formally your statement is the following: $\forall \mathbb{N}, \exists x : 1\cdot x = x$, such that $1 \in \mathbb{N}$..this is essentially the definition of the integers (although I only did it for the natural numbers).

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    $\begingroup$ This question comes from someone trying to understand how the concepts of factors and multiples apply in an extreme case. It is not helpful to tell someone their question is circular. If you don't understand what they meant to ask, don't answer. Your first paragraph might start to help, but needs to be expanded. $\endgroup$ – Sue VanHattum Oct 26 '14 at 14:41

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