### Mental Math Tricks: Is this divisible by 17?

So, most know how to tell if something is divisible by 2 or 5, and many know how to tell if something is divisible by 9. What about other numbers?

So, here are strategies for discovering divisibility from 2 to 10, and then we'll talk about some rarer, more surprising divisibility tricks:

### Divisible by two:

If the number ends in 0, 2, 4, 6, or 8, it is divisible by 2.

### Divisible by five:

If the number ends in 0 or 5, it is divisible by 5.

### Divisible by ten:

If the number ends in 0, it is divisible by 10.

### Divisible by nine:

If you add all the digits in a number together and that new number is divisible by 9, then it is also divisible by 9.

Example #1: 189 -> 1 + 8 + 9 = 18, 18 is divisible by 9, so 189 is also divisible by 9
Example #2:
137781 -> 1 + 3 + 7 + 7 + 8 + 1 = 27, 27 is divisible by 9, so 137781 is also divisible by 9
(Note: If adding six numbers together in your head seems difficult, look for my next post on number-grouping tricks. Soon, adding six numbers together will be a cinch!)

### Divisible by three:

If you add all the digits in a number together and that new number is divisible by 3, then it is also divisible by 3.

Example #1: 66 -> 6 + 6 = 12, 12 is divisible by 3, so 66 is also divisible by 3
Example #2:
15309 -> 1 + 5 + 3 + 9 = 18, 18 -> 1 + 8 = 9, 9 is divisible by 3, so 18 is divisible by 3; 18 is divisible by 3 so 15309 is divisible by 3

### Divisible by six:

If the number is divisible by 2 and 3, then it is divisible by 6.

Example #1: 72, 72 is divisible by 2 because it ends in a 2, 72 is divisible by 3 because 7 + 2 = 9 and 9 is divisible by 3, so 72 is also divisible by 6
Example #2: 30618, 30618 is divisible by 2 because it ends in an 8, 30618 is divisible by 3 because 3 + 0 + 6 + 1 + 8 = 18 and 18 is divisible by 3, so 30618 is also divisible by 6

### Divisible by four:

If the last two digits are divisible by 4, then it is also divisible by 4. Or, if you can cut it in half twice evenly.

Example #1: 124, 24 is divisible by 4 (cut 24 in half -> 12, then in half again -> 6), so 124 is divisible by 4
Example #2: 562312, 12 is divisible by 4, so 562312 is divisible by 4

### Divisible by eight:

If the last three digits are divisible by 8, then it is also divisible by 8. Or, if you can cut it in half evenly three times.

Example #1: 200, cut 200 in half -> 100, cut 100 in half -> 50, cut 50 in half -> 25, because 200 cuts in half evenly three times, it is divisible by 8
Example #2: 429192, 192 is divisible by 8 (192 in half is 96, 96 in half is 48, 48 in half is 24), so 429192 is divisible by 8

### Divisible by seven:

Take the last digit, double it, and subtract it from the rest. If that number is divisible by 7, then your original number is also divisible by 7.

Example #1: 266, take the last digit (6) and double it -> 12, subtract it from the first two digits -> 26 - 12 = 14. 14 is divisible by 7, so 266 is divisible by 7.
Example #2: 13034, take the last digit (4) and double it -> 8, subtract it from the rest of the digits -> 1303 - 8 = 1295. Is 1295 divisible by 7? Take the last digit (5), and double it -> 10, subtract it from the rest of the digits -> 129 - 10 = 119. Is 119 divisible by 7? Take the last digit (9) and double it -> 18, subtract it from the rest of the digits -> 11 - 18 = -7. -7 is divisible by 7, so 119, 1295, and 13034 are all divisible by 7. Phew!

## What's with the divisible by seven rule?

So, the divisible by seven rule is a unique one. For all you budding mathematicians, here is why it works.

Let's take a number n that has more than one digit.

That means that it can be written like this:
10*x + y where y is the last digit and x is all the others.
Example: 147 can be written as 10*14 + 7

Remember that we said that if n is divisible by 7, then the difference between the rest of the numbers and twice the last digit would be a multiple of seven. So x - 2*y should be a multiple of 7. Let's write that as 7*z. So x - 2*y = 7*z, or x = 7*z + 2*y.

Thus, we get:
10*(7*z + 2*y) + y
Example: 378 can be written as 10*(7*3 + 2*8) + 8

Almost there. Now, it's just algebra.

10*(7*z + 2*y) + y = 70*z + 20*y + y = 70*z + 21*y = 7*(10*z + 3*y)

And that's it! n = 7*(10*z + 3*y), which means that n is divisible by 7

## Interesting. Can that be done with other numbers? What does this have to do with 17?

Notice the reason that 7 could be factored out of the second term. It's because we had a multiple of 10 (20 in that case) added to 1, which created a number that was divisible by 7, in that case 21. Are there other multiples of ten that, when 1 is added to it, give us a composite number?

31 and 41 don't work, because they are prime. But, 51 does work! If we could have that multiple of 10 be 50, then we could probably come up with a new divisibility rule.

What is 51 divisible by? Well, it happens to be divisible by 17. Could we possibly get a 17 divisibility rule out of this?

Let's look at the pattern again. 7*(10*z + 3*y) is the end pattern that we are looking for. Let's replace 7 with 17. We get 17*(10*z + 3*y) => 170*z + 51*y => 170*z + 50*y + y => 10*(17*z + 5*y) + y

Does that look somewhat familiar? Check out the first equation that we got to represent the divisible by 7 rule: 10*(7*z + 2*y) + y. It looks pretty close.

So, let's try translating it back into English: If you multiply the last digit by 5 (instead of 2), and subtract it from the rest of the digits, if that number is divisible by 17, then the original number is divisible by 17.

Experiment #1: 867, multiply the last digit (7) by 5 -> 35, and subtract it from the rest of the digits -> 86 - 35 = 51. 51 is divisible by 17 (17*3), so 867 is divisible by 17.
Experiment #2: 6936, multiply the last digit (6) by 5 -> 30, and subtract it from the rest of the digits -> 693 - 30 = 663. Is 663 a multiple of 17? Multiply the last digit (3) by 5 -> 15, and subtract it from the rest of the digits -> 66 - 15 = 51. 51 is divisible by 17, so 663 and 6936 are also divisible by 17.

## Wow! Will it work for other numbers?

I think you probably know the answer to that. :) The other two that I found were 13 and 27. Can you figure out divisibility rules for those? Good luck! Okie said…
um...wow. my head is spinning. cool post though. thanks man. :)

### How Many Teeth Does The Tooth Fairy Pick Up Each Night in Utah?

Somebody asked me a question about my Tooth Fairy post the other day that got me thinking. How many baby teeth are lost every day in Utah?

I began with Googling. Surely someone else has thought of this and run some numbers, right? Lo, there is a tooth fairy site that claims that the Tooth Fairy collects 300,000 teeth per night.

That's a lot; however, when I ran the numbers, it started to seem awfully low.

Let's assume that the Tooth Fairy collects all baby teeth regardless of quality and we assume that all children lose all their baby teeth. The world population of children sits at 2.2 billion, with 74.2 million of them in the United States. Of those, approximately 896,961 of them are in Utah. This means that somewhere around .04077% of the world's children are in Utah.

If we assume that kids in Utah lose teeth at the same rate as all other children in the world and that each day in the year is just as likely as the rest to lose a tooth, then we have that of the alleged …

### BYU and the Sunday Compromise?

I read an article by Brad Rock this morning where he quoted heavily from Dr. Thomas Forsthoefel who was giving his opinion on religious institutions being involved in sports. BYU, of course, came up.
I think Forsthoefel came off sounding a bit misinformed about the culture, drive, mission, etc. of BYU. Below is the email that I sent to Brad Rock this morning after finishing the article:
That was an interesting article. I tend to disagree with Forsthoefel, though, or at least disagree with what I may have read into his comments.
A quote in your article says:
"There may be a kind of growing pain. BYU is in the real world and the real world works on Sunday. Can we (BYU) live with the adjustment? I'm empathetic with that, whatever decision is made, people are going to be unhappy.… Some will say get with the program, we'll be OK at the next level, others will say we've sold out and we've made a deal with the world."

This seems to suggest one or two things; fir…

### Baby Names: What my daughter's name has to do with an ancient Persian Fairy Tales

If you read my previous post on my sons' names, you'll know that this post is about my daughters' names.

When we found out that we were going to have twins, I vowed that there names were not going to rhyme or alliterate. We weren't going to do Jadyn and Jordan, or Kim and Tim, or Esther and Edgar (all likely candidates for other, less elitist parents, especially Esther and Edgar). I did want the names to have something to do with one another somehow.

Felicity Mae CummingsFelicity's first name has little to do with its underlying Hebrew meaning or its tie to Biblical history and everything to do with the fact that this was a name that Kristi had always wanted one of her girls to have because she liked that it meant "happiness".

So, to tell you the truth, I didn't do a lot of research on this name because its place in our family had already been decided.

But, it was excellent material to work with. The initial spark that 'Felicity' provided gave …