## More Curious Calculations

The next few calculator curiosities are variations on one basic theme.

(1) Enter a three digit number — say 471. Repeat it to get 471471.

Now, divide that number by 7, divide the result by 11, and d ivide the result by 13. Here we get:

471471/7=67353

67353/11=6123

6123/13=471

which is the number you first thought of.

Try it with other three digit numbers — you will find that exactly the same trick works.

Now, mathematics isn’t just about noticing curious things — it’s also important to find out why they happen. Here, we can do that by reversing the entire calculation. The reverse of division is multiplication, so — as you can check — the reverse procedure starts with the three digit result 471, and gives

$471 \times 13=6123$

$6123 \times 11=67353$

$67353 \times 7=471471$.

Not terribly helpful as it stands …but what this is telling us is that

$471 \times 13 \times 11 \times 7=471471$.

So, it could be a good idea to check what $13 \times 11 \times 7$ it s. Get your calculator and work this out. What do you notice? Does it explain the trick?

(2) Another thing mathematicians like to do is “generalize”. That is, they try to find related ideas that work in similar ways. Suppose we start iwth a four digit number, say 4715. What should we multiply it with to get 47514715? Can we achieve that in several stages, multiplying by a series of smaller numbers?

To get started, divide 47154715 by 4715.

(3) If your calculator runs to ten digits, what would be the corresponding trick with five digit numbers?

(4) If your calculator handles numbers with at least 12 digits, go back to a three digit number, say 471 again. This time, instead of multiplying it by 7, 11, and 13, try multiplying it with 7, 11, 13, then 101, then 9901. What happens? Why?

(5) Think of a three digit number, such as 128. Now multiply it repeatedly by 3,3,3,7,11,13 and 37. (Yes, three multiplications by 3). The result is 127999872 —- nothing special here. So, add the first number you thought of: 128. Now, what do you get?

Have fun!!

More later,

Nalin Pithwa

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