## Exploding Dots

### 6.8 Wild Explorations

Here are some “big question” investigations you might want to explore, or just think about. Have fun!

 EXPLORATION 1: CAN WE EXPLAIN AN ARITHMETIC TRICK?  Here’s an unusual way to divide by nine.   To compute $$21203 \div 9$$, say, read “$$21203$$” from left to right computing the partial sums of the digits along the way and then read off the answer $$21203\div 9 = 2355 \; R8$$. In the same way, $$1033 \div 9 = 114 \; R7$$ and $$2222 \div 9 = 246 \; R8$$.   Can you explain why this trick works?   Here’s the approach I might take: For the first example, draw a picture of $$21203$$ in a $$1 \leftarrow 10$$ machine, but think of nine as $$10-1$$. That is, look for copies of in the picture.

 EXPLORATION 2: CAN WE EXPLORE NUMBER THEORY? Use an $$1 \leftarrow x$$ machine to compute each of the following   a) $$\dfrac{x^{2}-1}{x-1}$$         b)  $$\dfrac{x^{3}-1}{x-1}$$      c) $$\dfrac{x^{6}-1}{x-1}$$      d) $$\dfrac{x^{10}-1}{x-1}$$   Can you now see that $$\dfrac{x^{number}-1}{x-1}$$ will always have a nice answer without a remainder?   Another way of saying this is that $$x^{number}-1 = \left(x-1\right) \times \left(something\right)$$.   For example, you might have seen from part c) that $$x^{6}-1 = \left(x-1\right)\left(x^{5}+x^{4}+x^{3}+x^2+x+1\right)$$. This means we can say, for example, that $$17^{6}-1$$ is sure to be a multiple of $$16$$! How? Just choose $$x=17$$ in this formula to get   $$17^{6}-1 = \left(17-1\right) \times \left(something\right)=16\times \left(something\right)$$.   a) Explain why $$999^{100}-1$$ must be a multiple of $$98$$.   b) Can you explain why $$2^{100}-1$$ must be a multiple of 3, and a multiple of 15, and a multiple of 31 and a multiple of 1023? (Hint: $$2^{100}=\left(2^2\right)^{50}=4^{50}$$, and so on.)   c) Is $$x^{number}-1$$  always a multiple of $$x+1$$? Sometimes, at least?   d) The number $$2^{100}+1$$ is not prime. It is a multiple of $$17$$. Can you see how to prove this?

 EXPLORATION 3: AN INFINITE ANSWER?  Here is a picture of the very simple polynomial $$1$$ and the polynomial $$1-x$$. Can you compute $$\dfrac{1}{1-x}$$? Can you interpret the answer? (We’ll explore this example in the next chapter.)

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