Question 21
The distinct prime factors of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$.
Let $A = 2^{k} - 2$ and $B = 2^{k} \cdot A$, where $k$ is an integer $(k \gt 1)$.
Show that, for every choice of $k$,
a. $A$ and $B$ have the same set of distinct prime factors.
b. $A+1$ and $B+1$ have the same set of distinct prime factors.
Let $A = 2^{k} - 2$ and $B = 2^{k} \cdot A$, where $k$ is an integer $(k \gt 1)$.
Show that, for every choice of $k$,
a. $A$ and $B$ have the same set of distinct prime factors.
b. $A+1$ and $B+1$ have the same set of distinct prime factors.