Consider $a= \langle a_0, \cdots ,a_{m−1} \rangle$ and $b = \langle b_0, \cdots ,b_{n−1} \rangle$ then the concatenation of a and b, written $a*b$, is $\langle a_0,\cdots ,a_{m−1},b_0,\cdots,b_{n−1} \rangle$. Show that the function that takes a and b to a ∗ b is representable in $T_E$ which is the theory axiomatized by $A_E$ or the axioms of arithmetic.
Here are my thoughts on how to represent this concatenation: I know I need to use prime factorizations but don't really understand how. I think I can represent $a$ by the product of $2^{a_0+1} \cdot \cdots \cdot prime_{m-1}^{a_{m-1}+1}$ and do the same with $b$. Then I need to figure out how to put these together and I can use divides and does not divide when it comes to primes.
Any tips would be much appreciated!