Quick - is 91 prime?

Starting from the left add every other digit => n1, Starting with the second digit add every other digit => n2

if n1 - n2 == 0 or if n1 - n2 is divisible by 11 then the number is divisible by 11. Example

13574

1 + 5 + 4 = 10

3 + 7 = 10

Since 10 - 10 == 0, the number is divisible by 11.

Even cooler example 155357548018579643

1 + 5 + 5 + 5 + 8 + 1 + 5 + 9 + 4 = 43

5 + 3 + 7 + 4 + 0 + 8 + 7 + 6 + 3 = 43

43 - 43 == 0, also divisible by 11

Here's some more fun: http://www.jgc.org/blog/2008/02/sum-of-first-n-odd-numbers-i...

http://www.spaceandgames.com/?p=27

Now how certain are you that 51 is prime?

for example, here's a quicker way for 1353 than what staticshock posted. 1353 - 1100 = 253. and 253 - 220 = 33. And clearly 11 divides 33, so 11 divides 1353.

modular arithmetic ftw!

you want to know if 8765 is divisible by 7. take the one's digit and multiply by 21. this gets a number that is divisible by 7 so you can subtract it. this will clear the last digit because we're subtracting this:

last digit * 1 + last digit * 20.

now we have a number that ends in a zero. in this case: 8660

now divide by 10. we can do that because that's dividing by 2 and 5 and that won't affect whether or not 7 can go into the number.

now we have a number to deal with that's at least one digit shorter. repeat until it's obvious if the remaining number divides by 7 or not.