There are divisors, and then there are divisors, and then there are still more divisors!

Suppose that I am tasked with dividing 15 by 6. What can be say about these numbers, and the result? In this task, 6 is called the divisor, and 15 is called the dividend, yes, as we all know. Regarding the result, we have the quotient (2, in this case), and the remainder (3, in this case). Well and good, but the terminology becomes tricky after this point. How tricky? Shibboleth-tricky. The terminology, concerning the word ‘divisor’, is, on the surface, not only counterintuitive, but if you don’t get it right, if you conflate the later meaning with this earlier meaning, you classify yourself as a babe in the woods regarding this domain of discourse (elementary Number Theory). So, what is the next, oh-so-tricky, sense of ‘divisor’? The mystery disappears if you have enough innate mathematical sensitivity / maturity to admit that, for a given number considered as a dividend, those divisors resulting in a REMAINDER OF 0 have, should be accorded, special status, and so they are. In common parlance they are called the ‘factors’ of the given number. So, to continue with our example, the divisors of 15 are 1, 3, 5, and 15. So, ‘divisor’ is being used here as an abbreviation for ‘divisor resulting in a remainder of 0’. With that, the mystery disappears, but there is still the question of why we don’t just use the term ‘factor’ instead. The answer to that is based on a certain undercurrent in mathematics education, namely, that in some respects the terminology used in elementary school differs from that used in secondary school. For example, in elementary school, a square is not considered to be a rectangle, but in secondary school it is. In a similar manner, ‘factor’ (the noun, not the verb) gets later replaced by ‘divisor’. Okay, so far so good, but wait, there’s more, namely, what happens when 0 is the dividend? For example, what is 0 divided by 7? The answer is 0, of course. So, 7 is a divisor of 0, right? Yeeeessss, as is in fact EVERY positive integer, but we say, in fact, that 0 HAS NO DIVISORS. How can this be? The answer is that it is the useful take on something mathematically significant that may or may not happen. To understand this, consider the number 21. Now 7 is a divisor of 21, right? The quotient is a NONZERO NUMBER, 3, and the remainder is 0. So, the divisor, 7, times that nonzero quotient, 3, equals the dividend, 21. Well and good, but does this kind of thing happen for 0? No, never. In our example, the quotient is 0 (not a nonzero number), and it is a triviality that the quotient time the dividend will be 0. We wish to EXCLUDE THIS TRIVIALITY from the notion of ‘divisor’, and we do so by employing the seemingly-contradictory phraseology that ‘there are no divisors of 0’, which, not to beat a dead horse, means ‘there are no positive integers p and q such that pq = 0. This linguistic hairball is the price we pay for excluding that triviality but, from the standpoint of mathematics, it’s worth it.

Whew, that is all on that, except for the lingering question about the negative numbers. For example, shouldn’t we say that the divisors of 15 are 1, 3, 5, 15, -1, -3, -5, and -15? Well, we COULD, but as is universally the case across all domains of discourse, we normally exclude mention of things whose mention brings no further information to the table. To give a somewhat more meaty example of this, suppose we wanted to compile a list of some Pythagorean triples, that is, triples of positive integers x, y, and z such that the square of x plus the square of y equals the square of z. The case of 3, 4, and 5 is well-known. Another case is 8, 15, and 17. Suppose that we have recorded those two cases, and next, somehow, comes to our attention the case 6, 8, and 10. What would you say? After a moment’s reflection, you might well say: “Hey, that’s not helping us any, since we already have the triple 3, 4, and 5 on the list. If a triple is already on the list, then multiplying the elements of the triple through by any given positive integer will, trivially, also be a Pythagorean triple, and so we don’t need to record such cases. In this case each of 3, 4, and 5 was multiplied by 2, but (2 x 3)^2 + (2 x 4)^2 = (2 x 5)^2 if, and only if, 4 x 3^2 + 4 x 4^2 = 4 x 5^2 if, and only if, 4(3^2 + 4^2) = 4 x 5^2 if, and only if, 3^2 + 4^2 = 5^2, our original equation. So, the concept of Pythagorean triples provides another example of the simplifying suppression of useless results. This could segue into a discussion of equivalence relations, but we’ll leave it at that, but remember that with increasing complexity of topic comes the necessity of an ever-increasing set of ‘unstated conditions’ so that discussion does not become unwieldly.

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