Find the total number of identical terms in 2 sequences.

Question

I am having difficulty understanding the solution. What is k? Please explain how is this solved using k :(

Answer 1

Note that $p+1=7\cdot\frac{q}{4}$. Since $p$ is a natural number, and $4$ and $7$ are coprime, implies, $q$ must be divisible by $4$. By similar arguments, one can say that $p+1$ is divisible by $7$.

In fact, you get: $$\frac{p+1}{7}=\frac{q}{4}$$ which hints at the fact that the result of dividing $p+1$ by $7$, or dividing $q$ by $4$ will yield the same natural number (since they both are divisible). $k$ is that same natural number.

Once you arrive at $p+1=7k$ and $q=4k$, you can plug those values into the inequalities as the book has done. Since $k$ is a natural number, at the last step, you're able to narrow down the number of possible valid solutions from $k\leq93/7$ to simply $k=1,2,3,...,13$. This is the role of $k$.

Answer 2

As $q$ is an integer, $7$ must divide $4(p+1)$

Hence $7|(p+1)$ as $(4,7)=1$

$\implies p +1$ can be written as $7k$ where $k$ is any integer