Mathematicians have the same definition of factor as what you used. It is your definition of repeated factor that I was confused about. To me if f divides n, and f divides n/f, then f is a repeated factor of n. It seems from your description that to you say that f is a repeated factor of n if and only if f*f=n.
That isn't confusing in a "non-mathematician vs mathematician" way. That is confusing in a "you used a phrase that described what you meant without noticing that it was very ambiguous" way. That is, I am confident that I would not have understood that as you intended back in highschool. And I was likewise not confused when I first saw the phrase "this factor is repeated 3 times".
In ths case your reasoning has a non-obvious step from "odd number of factors" to "one factor must be the square root of the number". It is easy enough to fill that in - just add a comment that you can pair off factors by putting i with j if i*j=n, and you will have a left-over if and only if there is a factor i with i*i=n. But since that pairing is the key step, it needs to actually be stated. You can't expect other people to have that flash of brilliance. | [reply] |
I've seen math texts for beginners that present factors as occuring pairs. For instance, 12 can be factored 1*12=12, 2*6=12, 3*4=12,
giving factors 1,2,3,4,6,12. Perhaps this is the source of this use of "repeated"; a number such as 16 is factored 1*16=16, 2*8=16, 4*4=16, giving "factors" 1,2,4,4,8,16.
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