please explain clear show steps Solution 5. If a|b and b|c, then there exist integers p and q such that b = ap and c = bq. So c = bq = (ap)q = a(pq), which is a times an integer. So a|c. 6. If a|c then c = an for some integer n. Then ck = (an)k = aknk = a(ak-1nk), which is a times an integer (because a,k, and n are all integers so ak-1 and nk are integers). So a|ck. 7. a|b and b|c, so a|c, which implies a|ck for any positive integer k..

1. please explain clear show steps
Solution
5. If a|b and b|c, then there exist integers p and q such that b = ap and c = bq. So c = bq = (ap)q =
a(pq), which is a times an integer. So a|c.
6. If a|c then c = an for some integer n. Then ck = (an)k = aknk = a(ak-1nk), which is a times an
integer (because a,k, and n are all integers so ak-1 and nk are integers). So a|ck.
7. a|b and b|c, so a|c, which implies a|ck for any positive integer k.