Sunday Afternoon Maths XXI

Here's this week's collection. Answers & extensions here. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit. This Tuesday is Maths Jam, make sure you're there!

Sum Equals Product

Source: The Penguin Book of Curious and Interesting Puzzles by David Wells

$3$ and $1.5$ are a special pair of numbers, as $3+1.5=4.5$ and $3\times 1.5=4.5$ so $3+1.5=3\times 1.5$.

Given a number $a$, can you find a number $b$ such that $a+b=a\times b$?

Wool Circles

$n$ people stand in a circle. The first person takes a ball of wool, holds the end and passes the ball to his right, missing $a$ people. Each person who receives the wool holds it and passes the ball on to their right, missing $a$ people. Once the ball returns to the first person, a different coloured ball of wool is given to someone who isn't holding anything and the process is repeated. This is done until everyone is holding wool.

For example, if $n=10$ and $a=3$:

In this example, two different coloured balls of wool are needed.

In terms of $n$ and $a$, how many different coloured balls of wool are needed?