If p is a prime and gcda b p What are the possible values o

If p is a prime and gcd(a, b) = p, What are the possible values of gcd(a^2, b)? gcd(a^3, b)?

Solution

gcd(a,b)=p means p is the only prime that divides both and b

a)

p|a hence, p^2|a^2

p|b but it is possible that a higher power of p that 1 divides b. eg. p^2,p^3 and so on.

Sincd gcd(a,b)=p it is true that one of them is divided by p and no higher power of p. If it is b then

gcd(a^2,b)=p

If it is a. Then,

gcd(a^2,b)=p^2

a)

p|a hence, p^3|a^3

p|b but it is possible that a higher power of p that 1 divides b. eg. p^2,p^3 and so on.

Sincd gcd(a,b)=p it is true that one of them is divided by p and no higher power of p. If it is b then

gcd(a^3,b)=p

If it is a. Then,

gcd(a^3,b)=p^3