Since this is the last post of 2012, I thought I'd finish the year with a mental math technique. Before I do that though, I want to show you a pattern that is closely related.
Take any three digit number. For this, I will use 123.
Multiply that by 13. You should get 1599.
Now, multiply that by 11. You should get 17589. You might have done that in your head using the multiplication by eleven trick I taught last October.
Now, multiply that by 7. This will give you 123123. And if you'll notice, we started with the number 123 and finished with two 123s.
Why did this happen? It is very simple, and just relies on the fact that 13 x 11 x 7 is 1001. So, by multiplying by these seemingly random numbers is really multiplying by 1001. And any number times 1001 is just itself repeated twice.
Division by 1001 results in a similar answer. For instance, 123 ÷ 1001 = 0.122877122877...
If you notice, the first three digits are 122, which is just 123 - 1. The next three are 877, and 122 + 877 = 999.
This pattern continues as well. I am not sure how to prove that, but please comment if you do know.
So, to divide by 1001, you just subtract one for the first three digits, and subtract the first three digits from 999 for the next three digits.
However, telling someone you can divide any number by 1001 doesn't sound that impressive. Since 1001 is right next to 1000, people will get very suspicious.
That is why I showed you the multiplication pattern. Dividing by 1001 is basically dividing by 13, then 7, then 11.
123 ÷ 1001 = 123 ÷ (13 x 7 x 11)
So, instead of going all the way to 1001, let's just get part way there by dividing by the 13 x 7, which is 91. We will use 33 as the number we are dividing by, or the dividend.
Our goal is to make it a number divided by 1001, since we know how to do that. That means that the first step is multiplying the 91 by eleven. But, if we multiply the denominator by eleven, we must also multiply the numerator by eleven.
33 x 11
91 x 11
We know 91 x 11 is 1001, so we have the problem we are looking for. What is 33 x 11 though? If you go to the multiplication by eleven post, you will see that it is just 363 (add the 3 + 3, and stick it in the middle). This gives us the problem 363 ÷ 1001.
363 - 1 is 362 and 999 - 362 is 637, so the answer is 0.362637362637...
So, to divide by 91, you just multiply the number by 11, subtract one, and subtract that from 999 to get the answer.
Though it is a lot tougher, you can also divide by 77 (11 x 7) and 143 (13 x 11) with the same principle. You are just multiplying the top number by 13 or 7 instead of the easy 11.
The 1001 pattern is really cool, and something you can turn into a really cool trick. The multiplication by eleven is also a really cool effect. But putting them together into the division by 91 is a really impressive feat of mental math that will fool all of your friends.