let n=p1^r1.p2^r2.p3^r3...pk^rk where p1,p2,..pk are distinct primes and each ri>=0. prove that n is a perfect square if and only if each ri is even.
Hi, lenin1234007.
If each exponent is even then you can write
$\displaystyle n=(p_{1}^{l_{1}}\ldots p_{k}^{l_{k}})^{2}$
Where $\displaystyle l_{1}=r_{1}/2, \ldots, l_{k}=r_{k}/2$.
If n is a perfect square, then
$\displaystyle n=t^{2}$
This implies
$\displaystyle p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}=t^{2}$
Now t has its own unique factorization into primes. By the Fundamental Theorem of Arithmetic, the primes and their exponents must be equal on both sides of the equation. Since the exponents of all the primes on the right are even after you square, all the exponents on the left must be even also.
Does this help? Let me know if something is unclear.
Good luck!