Show that if a | b and b | a, then either a=b or a=-b ?

Question

show that if a | b and b | a, then either a=b or a=-b?

Answer 1

The following trivial sub-theorem is needed, which is

proved by contradiction:

For two integers x and y, if xy=1 then x=y=1 or x=y=-1.

Proof:

then y = 1/x for x neither 1 nor -1 nor zero

then y is a proper fraction, a contradiction

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Main theorem: integers a and b such that a|b and b|a;

then a=b or a=-b

Note that neither a nor b can be zero, since they are divisors!

since a|b, a = nb for some integer n.

since b|a, b = ma for some integer m.

Then

a = nb = n(ma) = (nm)a <-- substitution/associative

so a = (nm)a by transitive/substitution

1 = nm <--- divides by non zero a

since n and m are integers, n=m=1 or n=m=-1

by supporting theorem.

then a = bm = b(1) = b or a=bm = b(-1) = -b

[end of proof]