Here is an addition problem in a \(1 \leftarrow 5\) machine. (That is, it is a problem in base five.) This is not a \(1 \leftarrow 10\) machine addition.

a) What is the \(1 \leftarrow 5\) machine answer?

b) What number has code \(20413\) in a \(1 \leftarrow 5\) machine? What number has code \(13244\) in a \(1 \leftarrow 5\) machine? What is the sum of those two numbers and what is the code for that sum in a \(1 \leftarrow 5\) machine?

[Here are the answers so that you can check your clever thinking.

The sum, as a \(1 \leftarrow 5\) machine problem, is

\(20413+13244 = 3|3|6|5|7 = 3|4|1|5|7 = 3|4|2|0|7 = 3|4|2|1|2 = 34212\).

In a \(1 \leftarrow 5\) machine, \(20413\) is two \(625\)s, four \(25\)s, one \(5\), and three \(1\)s, and so is the number \(1358\) in base ten; \(13244\) is the number \(1074\) in base ten; and \(34212\) is the number \(2432\) in base ten.  We have just worked out \(1358 + 1074 = 2432\).]

 Let’s work with a \(1 \leftarrow 3\) machine.

a) Find \(111 \times 3\) as a base three problem. Also, what are \(1202 \times 3\) and \(2002 \times 3\)?

     Can you explain what you notice?

Let’s now work with a \(1 \leftarrow 4\) machine.

b) What is \(133 \times 4\) as a base four problem? What is \(2011 \times 4\)? What is \(22 \times 4\)?

     Can you explain what you notice?

In general, if we are working with a \(1 \leftarrow b\)  machine, can you explain why multiplying a number in base \(b\) by \(b\) returns the original number with a zero tacked on to its right?