Assume we have integers a, b, and c such that c =ab and gcd(a,b)=1. Show that c is a perfect square if and only if a and b are perfect squares.

Assume we have integers a, b, and c such that c =ab and gcd(a,b)=1. Show that c is a perfect square if and only if a and b are perfect squares.
This can be done using the prime factorization of the numbers a and b. There might be a faster way, though.

-Dan

Let $a=p_1^{a_1}...p_n^{a_n}$ and $b=q_1^{b_1}...q_m^{a_m}$ so if $ab = p_1^{a_1}...q_m^{b_m}$ is a square it means all exponents $a_1,a_2,...,b_m$ are even so $a$ and $b$ have even exponents and so are squares.

5. Originally Posted by ThePerfectHacker
Let $a=p_1^{a_1}...p_n^{a_n}$ and $b=q_1^{b_1}...q_m^{a_m}$ so if $ab = p_1^{a_1}...q_m^{b_m}$ is a square it means all exponents $a_1,a_2,...,b_m$ are even so $a$ and $b$ have even exponents and so are squares.
That wasn't so long a proof. Though I'm sure I wouldn't have been able to state it so succinctly.

-Dan

6. Originally Posted by topsquark
That wasn't so long a proof. Though I'm sure I wouldn't have been able to state it so succinctly.
Thank you for the +rep+ points my reputation explode up so fast.

7. Originally Posted by ThePerfectHacker
Thank you for the +rep+ points my reputation explode up so fast.
Really? How many points did you jump? (I didn't know my Kung Foo was that strong. )

-Dan

8. Originally Posted by topsquark
Really? How many points did you jump? (I didn't know my Kung Foo was that strong.
I am not even sure I think something like 50.

9. Originally Posted by ThePerfectHacker
I am not even sure I think something like 50.
Criminy! If I can do that, imagine what Jhevon could do.

-Dan