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I think you missed the (implicit) definition of “the sum”: If someone asked you “what is the sum of all the proper divisors of 8?”, would you say it's 1+2+4=7, or would you say, “well, it could be either 1+2+4=7, or 1+2+2+4=9, or 1+1+1+2+2+4+4+4=19, or…”? Anyway, yes, reading such definitions comes with experience (what's called “mathematical maturity”: https://en.wikipedia.org/wiki/Mathematical_maturity) — you have to assume that everything being stated is well-defined and makes sense, that every word matters (e.g. you can't ignore the word “all” in “the sum of all proper divisors”, nor can you ignore the word “the”!), and then, if that doesn't work, try to see if alternative definitions make sense. Also, sometimes things are stated in two different ways; that's usually helpful. In this case, given that the first two sentences are: > An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. Note the “That is”: If the first way of stating the definition doesn't seem clear / doesn't seem to fit the example that immediately follows, then try the second sentence, specifically follow the “aliquot sum” link to https://en.wikipedia.org/w/index.php?title=Aliquot_sum&oldid... which makes it very clear: > For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6). […] > The untouchable numbers are the numbers that are not the aliquot sum of any other number. |