We ought never to allow ourselves to be persuaded of the truth of anything unless on the evidence of our own reason – René Descartes
Prove that b | a iff every multiple of a is multiple of b
Prove that every multiple of some integer b is a multiple of another integer a
if, and only if b divides a.
This was one of homework problems in
Introduction to Abstract Algebra
I was in for two lectures.
Suppose , then for some integer k by the definition of divisibility.
Let so that for some integer m by the definition of .
Then by substitution.
Hence by associative law.
Thus as desired since .
Suppose for every e in , e is also in .
By the definition of , for some integer k for every e in .
Similarly, for some integer m since is a subset of .
Then .
In particular, if we choose , then for some integer n.
Hence as desired.
Since 1 and 2 are both proven, we have shown that if and only if .
Q.E.D.