When the positive integer k is divided by 7, the remainder is 6. What is the remainder when k+2 is divided by 7?
If m divided by n gives the quotient q and the remainder r, this means that m = qn + r and 0 <= r < n. Therefore, if m divided by n gives the remainder r, this means that for some integer q, m = qn + r and 0 <= r < n. Make sure you understand this and feel free to ask questions if you don't.
Now rewrite the phrase "When the positive integer k is divided by 7, the remainder is 6" using the pattern above and see what this says about k + 2.
Plato gave these numbers just as examples. I think it is instructive to answer his questions.
Yes, for some integer q we have k = 7q + 6. Therefore, k + 2 = 7q + 8. Can you represent this number as 7q' + r for some new q' and r such that 0 <= r < 7?
When the positive integer k is divided by 7, the remainder is 6. This means that there exists number n such that $\displaystyle k=7n+6$ holds.
Add 2 to both sides of that equation and you get:
$\displaystyle \begin{align*}k+2&=7n+6+2\\ k+2&=7n+8\\k+2&=7n+7+1\\ k+2&=7(n+1)+1 \end{align*}$
Now you see that the remainder must be 1.