Give a proof or counterexample, whichever is appropriate.

1. For integers a, b and c, ab|c only if a|c and b|c.

2.There is a unique integer b such that a|b for all integers a.

3. There is a unique integer a such that a|b for all integers b.

4. For all integers a, b, d, and q, if d|a and d|b, then d|aqb

5. For all integers a, b, d, and q, if d|b and d|aqb, then d|a

6. For all integers a, b, d, and q, {d Z : d|a AND d|b} = {d Z : d|b AND d|a qb}

7. For integers a and b, if |b| < |a| and a|b, then b=0