Let $a,a_1,a_2$ be integers and $b,b_1,b_2$ be odd positive integers.
Prove $(a_1a_2 / b) = (a_1/b)(a_2/b)$
Prove $(a / b_1b_2) = (a/b_1)(a/b_2)$ I need to prove these 2, i know both deal with the jacobi symbol but stuck.
Let $a,a_1,a_2$ be integers and $b,b_1,b_2$ be odd positive integers.
Prove $(a_1a_2 / b) = (a_1/b)(a_2/b)$
Prove $(a / b_1b_2) = (a/b_1)(a/b_2)$ I need to prove these 2, i know both deal with the jacobi symbol but stuck.
We change names slightly, for no very good reason. And again for no good reason, we do the second question first.
Recall that if $m$ is odd, and has prime power factorization $p_1^{e_1}\cdots p_k^{e_k}$, then by the definition of the Jacobi symbol, $(x/m)=\prod_{i=1}^k (x/p_i)^{e_i}.\tag{$1$}$ (The symbols on the right are Legendre symbols. Unfortunately, the same notation is used for each.)
The formula remains correct if we include in the above product some primes $p$ that actually don't divide $m$. For $(x/p)^0=1$, so does not affect the above product.
Now consider $(x/bc)$. Let $b=\prod_{i=1}^k p_i^{e_i}$ and $c=\prod_{i=1}^k p_i^{f_i}$. (Remember that some $e_i$ or $f_i$ can be $0$.) Then $bc=\prod_{i=1}^k p_i^{e_i+f_i},$ and now the result follows immediately from $(1)$.
For the first question, again use the basic defining Equation $(1)$. We want to show that $(x/m)(y/m)=(xy,m)$. To compute $(x/m)$, we find a product of terms $(x/p_i)^{e_i}$, where the $(x/p_i)$ are Legendre symbols. We compute $(y/m)$ and $(xy/m)$ is a similar way.
However, for Legendre symbols, the relationship $(xy/p)=(x/p)(y/p)$ holds. I am reasonably sure that this has already been done in your course. The result now follows immediately from $(1)$.