## Exploding Dots

### 2.5 ADDENDUM: Some Fun Extra $$1 \leftarrow 10$$ Questions

Lesson materials located below the video overview.

 Question 50:  MULTIPLYING BY TEN   Quickly compute each of the following: a) $$263207 \times 3$$ b) $$563872 \times 9$$ c) $$673600023 \times 2$$   d) Use the $$1 \leftarrow 10$$ machine to explain why multiplying a number in base $$10$$ by $$10$$ results in simply placing a zero at the end of the number.   e) Comment on the effect of multiplying a number written in base $$b$$ by $$b$$.   Comment: One can go much further with this thinking. What is the effect of dividing a number written in decimal notation by ten? By one-hundred?

 Question 51: MULTIPLICATION BY ELEVEN   Here’s a trick for multiplying two-digit numbers by $$11$$: To compute $$14 \times 11$$, say, split the $$1$$ and the $$4$$ and write their sum, $$5$$, in between: $$14 \times 11=154$$. To compute $$71 \times 11$$ split the $$7$$ and the $$1$$ and write their sum, $$8$$, in between: $$71 \times 11 = 781$$. In the same way: $$20 \times 11 = 220$$ $$13 \times 11 = 143$$ $$44 \times 11 = 484$$   It also works if we do not carry digits:   $$67 \times 11 = 6|13|7 = 737$$ $$48 \times 11 = 4|12|8 = 528$$     a)    Why does this trick work? b)    Quickly, what’s $$693 \div 11$$ ? c)     Quickly work out $$133331 \times 11$$.   A number is a palindrome if it reads the same way forwards as it does backwards. For example, 124454421 is a palindrome.   d)    TRUE OR FALSE and WHY: Multiplying a palindrome by $$11$$ produces another palindrome.

Question 52: AN UNUSUAL WAY TO DIVIDE BY NINE

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:

### $$21203 \div 9 = 2355 R 8$$.

In the same way,

$$1033 \div 9 = 1 | 1+0 | 1 + 0 + 3 | R 1 + 0 + 3 + 3 = 114 R 7$$

and

$$222 \div 9 = 246 R 8$$.

Why does this trick work? (Is the act of “carrying” digits a problem?)

HINT: Does it help to think of division by nine as division by “ten plus negative one”?

Comment: Alternatively … What is $$\frac{1}{9}$$ as a decimal? What do you obtain if you multiply $$21203$$ by this decimal?

 Question 53:  IS IT PRIME?    Is $$18^{1000} – 1$$  prime?

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