One of the great and simple remaining mathematical mysteries is the way that the primes are arranged. No one has yet determined formulas that even provide information on the density of primes (no matter what this bear says.)

So when looking at a natural number, it is critical to determine what primes divide into it. Which brings us to you. Everyone should know whether any number is divisible (when I say "divisible" I mean "divisible without remainder" as do most mathematicians) by any number 1-10. Here are the shortcuts:

Everything is divisible by 1 (1 is not prime by the way)

2, 5 and 10 are the ending tests. If it's divisible by 2 it ends with 2, 4, 6, 8 or 0. If it's divisible by 5 it ends with 5 or 0. And if it's divisible by 10 it ends with 0.

4 and 8 are slightly constructed. If the last two digits are divisible by 4, the the entire number is divisible by 4. The same holds for the last three digits and 8. If this is too hard, simply divide the last three digits in half and then check it is still divisible by 4.

3 and 9's are fun. Add all the individual numbers of the number together and if they are then divisible by 3, the number is divisible by 3. If they are divisible by 9, then the number is divisible by 9.

For 6, simply check if the number is divisible by 2 AND 3. If it is, then it is divisible by 6.

7... what a stupid number. It took me years to learn the sevens tests, and then (because I discovered it on my own without the help of a computer) I had to prove. I shan't prove it for you (the math isn't interesting enough) but I shall state the conclusion. To find if a number is divisible by 7 simply take the digit in the ones place and double it. Then subtract it from the remaining number. Continue doing this until you have a number you know is divisible by 7.

Example:

8,675,309

9x2=18 867,530-18 = 867,512

2x2=4 86,751-4 = 86,747

7x2=14 8,674-14=8,660

Zeros can simply be removed.

6x2=12 86-12 = 74 which is between 7x10 and 7x11.

So 8,675,309 is not divisible by seven.

Also of note:

My dad taught me the wheel of 7s for division.

Draw a circle. Start at the top and place these numbers clockwise around the circle evenly: 1, 4, 2, 8, 5, 7. Then, inside the circle, write these numbers respectively: 1/7, 3/7, 2/7, 6/7, 4/7, 5/7. Now to create a decimal representation of any fraction with 7 in the denominator, simply start at the number of sevenths left over on the wheel and proceed around it clockwise.

1/7 = .142857142857...

5/7 = .714285714285...

Two frightening uses of 7's in your everyday life. Think about it, won't you.