 On Thursday at the Good Dog, the 2nd round was multiply the numbers 11-20 by the number 77. One of the people playing was David, a math teacher and enthusiast. If any of you out there are math fiends, you’re gonna love the properties of the number 11 Dave was kind enough to share with us:

Thursday’s problem, “Provide every multiple of 77 between 77×11 and 77×20,” becomes a cinch when you know a simple math trick involving multiplying by 11.

*Multiplying By 11*
Take any number (for convenience’ sake, let’s deal with 813, but we could really talk about ANY number of ANY length). In order to multiply it by 11, all we need to do is write down the first number; add the last two numbers; add the next two numbers from the end; and write down the last number. In other words, 813×11=8[8+1][1+3]3, or 8943. 42×11=4[4+2]2, or 462. Etc. The only catch is remembering to carry the 1 if you get a two-number sum greater than 10: 491×11=4[4+9][9+1]1=[4+1][3+1]01=5401. Thus 77×11= 7[7+7]7=847.

More 11 weirdness after the jump…

Practice on your own: 51×11, 417×11, 39821×11

Back to the problem that was presented (“every multiple of 77 between 77×11 and 77×20″). Since 77 is 7×11, we can factor 11 out of every number and reformulate the question as 11x”every multiple of 7 between 7×11 and 7×20,” or 11×77, 11×84, 11×91, … 11×140. Adding 7 to each new multiple of 7 is a snap; multiplying the resulting numbers is also a piece of cake now that we have our handy-dandy shortcut. (77×11=847, 84×11=924, 91×11=1001, etc.)

*Verifying That A Number Is A Multiple Of 11*
“But wait,” there are those who must be asking. “What if I forget to carry the 1 or make a transcriptional error? If only there were a fast and easy way to figure out whether or not a number is divisible by 11!”

There is. Starting from the rightmost digit and working your way backwards, add up alternating digits in the number, subtract your two sums from each other, and if the answer is 0 or a multiple of 11, you’re all set. Let’s practice in order to solidify the skill.

Take any number (4708 is a nice one). In order to figure out whether or not it is evenly divisible by 11, we start counting from the end of the number (at the right side) and group every other digit together. In this case, we get {8,7} (last and third-to-last digits) and {0,4} (second-to-last and fourth-to-last digit). Adding up our groupings, we get 8+7=15 and 0+4=4. Subtracting the two sums, we get 15-4=11, which is a multiple of 11. 4708 is divisible by 11 (in fact, it’s 11×428).

Let’s try another number: 38742. Working backwards and counting every other digit, we write {2,7,3} and {8,4}, with respective sums of 12 and 12. Subtracting those gives us 0. 38742 is also divisible by 11 (38742=11×3522).

So it’s easy to go back and check our answers to the original problem. Although we can’t divide them quickly by 77 or larger numbers, we can check to see if they’re at least reasonable (i.e. divisible by 11, which is one of the factors of 77). If, for example, we accidentally wrote down 747 instead of 847 for 77×11, once we went back and checked we would get {7,7} and {4}, with sums of 14 and 4, which are not separated by 0 or a multiple of 11. We would know that the original multiplication was wrong, and it should be a fairly straightforward process to reference the original number and to locate the source of our mistake.

Verify for yourself whether the following numbers are divisible by 11: 8910440, 134002, 913, 385, 9712967

What about if a number isn’t divisible by 11?

*Figuring Out The Remainder When A Number Is Divided By 11*
Say that you and 10 friends decide to go out for drinks, and then they stick you with figuring out the bill because you’ve been showing off your fancy new math skills. You go to a really fancy new club and rack up a bill of \$747. We’ve just verified that \$747 is not divisible by 11, but how much is left over when everyone puts in his or her share?

It is in fact really easy to discover the remainder. Just use the very same trick from above–making sure to start with the last digit and to work your way to the left. For our \$747 bill, we count the digits and get {7,7} and {4}, which have a difference of 14-4=10. In other words, if you want to get back at your buddies for forcing you to calculate their contributions, you can make all 10 of them pay an extra buck, and you’ll wind up with exactly the right amount. You put in \$67, they put in \$68, and everyone’s happy. Especially you, because you can do mathematical gymnastics.

The final trick when it comes to this remainder thing is to remember that if you get an answer that’s 11 or higher, you need to keep subtracting 11 until you get a number between 0 and 10. If you get a negative number, you need to add 11 in order to get your remainder in the 0-10 range.

Now you give it a try: 123456, 82, 19103, 2009, 20009

*What’s That In Decimals?*
“All right, Mr. Math Smarty Pants,” you’re saying. “But what if my friends get ahold of this e-mail and learn how to figure out whether a number divides 11 evenly? How can I split up the bill as evenly as possible into dollars _and_ cents?”

Never fear. Once you’ve figured out your remainder using the above process, you’ve still got one trick up your sleeve: multiply the remainder by 9. Actually, multiply the remainder by 9, and write down the resulting two digit number. Repeatedly.

See, 1/11=0.0909090909…, 2/11=0.1818181818…, 3/11=0.2727272727…, and so on. Just don’t get fooled into thinking that 11/11=0.9999999… It is…but that turns out to be the same as 1.

So you can very quickly come up with the number of cents each person owes, within a cent or so, by multiplying your remainder by 9 cents. In the case of our \$747 bill, the remainder is 14-4=10, so each person pays 10×9=90 cents extra. 11x\$67.90=\$746.90, so we’re off by 10 cents. Force your friends to pay it; you’ve earned that much.

Your turn to divide up the bill 11 ways, accurate to within a cent each: \$349, \$115, \$11802, \$36