Only 0 divides 0
0 divides only 0 , perhaps? WFL On 2/12/15, W. Edwin Clark <wclark@mail.usf.edu> wrote:
Ask Mathematica to find the gcd of 0 and a or 0 and 0. I don't have Mathematica but I'll bet it gives gcd(0,0) = 0 and gcd(0,a) = a for any positive integer a.
A common definition of gcd(a,b) is that it is an integer d which divides a and b and for which there are integers x and y such that d = xa+yb. Thus if a = b = 0 then d = 0 and no other integer will do for d.
For a nice treatment of divisibility see Tom Apostol's treatment of divisibility in his book Introduction to Analytic Number Theory. A beautifully written book --- written after 25 years of teaching the subject.
As Apostol says: For integers a, b Definition: a | b if and only if b = ac for some integer c.
Thus Every integer divides 0 Only 0 divides 0
On Wed, Feb 11, 2015 at 6:51 PM, James Propp <jamespropp@gmail.com> wrote:
Mathematica refuses to answer the question, and I've seen textbooks that duck the issue as well ("Suppose *a* and *b* are integers, with *a* nonzero. We say *a* divides *b* if and only if ...").
In fact, Mathematica also refuses to answer the question "Does 0 divide 1?"
Is it a standard convention that *a*-divides-*b* is a relation on (*Z*\{0}) x *Z*, so that asking whether 0 divides 1 is no more sensible than asking whether pi divides the square root of 2?
I would've been naively inclined to the view that 0 divides 0 is TRUE, while 0 divides 1 is FALSE.
I'm afraid to post this question to MathOverflow, lest I be reprimanded for asking such an inappropriate question ("Is this a homework problem that your professor assigned you?"). You guys are nicer, and more to the point, you all know me.
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