Show that if the square of no primes divides b and a2 bc2 th

Show that if the square of no primes divides b and a^2| bc^2 the aic.

Solution

Case 1:a=1

Then 1|c for all integers c

Case 2: a>1

So a has at least one prime factor,say p.

So, a^2 is a multiple of p^2

But square of no prime divides b

So, p^2 does not divide b

But, p^2 divides bc^2

So, p |c^2

But, p is prime. So, p|c

But p was any prime factor of a.

HEnce any prime factor of a divides c

Hence, a|c