The order of any element of the multiplicative group $\mathbb{Z}_{43}^*$
There are 42 elements of the multiplicative group, so any element will have order 2, 3, 6, 7, 14, 21, or 42. So, any element $g$ such that $g^6\neq 1$ but $g^7=1$ has order 7. Then it is just a matter of trial and error.