Prove or disprove – If a divides b and b divides a does a=b

Prove or disprove: If a, b belong to the set of positive integers, and if a divides b and b divides a, then a=b. Does this hold if if a,b are not necessarily positive? Why or Why not?

Here is what I have:

If a and b are integers, we say a divides b if there exists an integer k so that b=ak. Thus:


Since a divides b and b divides a, we know that k,j must both be integer, which means k and j must both equal 1 or -1. This means:


This proves that a=b when we assume that a,b are both positive integers.

For the second part of the question, I am asked if I can still say a=b if the assumption about a, b is relaxed so that a,b belong to the integers (not the positive integers.) I think The answer is no, I cannot say that a=b for every case when we allow a,b to also be negative integers.

Here is my reasoning (I use the same idea that a=bj and b=ak here)


So $ |a|=|b|$ but not $a=b$

I’m very new to proofs so any suggestions or thoughts are greatly appreciated.

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Since $a,b \in \mathbb{Z^+}$, $a=b$ most definitely, as you have proved. You don’t have to worry about negatives here if it is given that $a,b$ are positive integers. If $a,b \in \mathbb{Z}$, then a counterexample would be $a=−6,b=6$. Hence, unless $a,b$ are strictly positive, the statement is not true.

Really, if $a/ b & b/a$ then we know $a=b$ because if the values of $a$ and $b$ are same then they will divide completely each other so, Their values will be either positive or negative.

So, $a=\pm b$.

Hence proved