In Apostol's analytic number theory book, one of the properties listed for divisibility is:
n | 0 (every integer divides zero)
and also, something like 'zero divides only zero'
Could someone please explain what the crap he means? I thought you couldn't divide by zero!
The thing is that he is looking at "divisibility", the definition of divisibility is that a|b means that there exists an integer c such that ac=b.
So 0|0 means there exists a c such that 0*c=0, does there exist such an integer? of course, in fact any will do. Note we can define "divisibility" without actually doing any division so no we are not dividing by 0.
Thanks kamil, I just realised I actually misread the original question lol.
n | 0 (n divides 0) is NOT n/0. It is the other way around (0/n)
n/0 is n
divided by 0, 0/n is n dividing 0. The two are opposite. So n | 0 is the same as saying 0/n = an integer.
For example, 1 | 5 because 5/1 = 5 an integer. 2 | 6 because 6/2=3 an integer.
Another example, you may come across a property:
if a | b, and b | c, then a | c. This means if a divides b, and b divides c, then a divides c. Or, b is able to be divided by a, and c is able to be divided by b. Then c is able to be divided by a.
proof: if a | b, then b = an for some n (so b/a = an/a = n integer) and b | c means c = bm for some m for the same reason.
So c = bm = (an)m = a(nm), so c/a = a(nm)/a = nm integer. Which means a | c.