A lot of people will be surprised when I can't do some base calculations quickly in my head, and then will be equally surprised when I can do others almost instantly. The fact is that lots of mathy people aren't that much more skilled at... errr... numbers, than most, but rather are just more familiar with them. I know you're never supposed to reveal your tricks, but this is just a bit of insight into how certain problems have quick shortcuts.

Example:

15*39. On the surface this looks like any old two digit multiplication. But to a math major who is used to factoring and such, it instantly becomes 15*(40-1) = 4*15 *10 - 15 = 600 - 15 = 585. Each step is very simple and can be done very fast, but you don't necessarily see the shortcut if you don't work with expressions often.

Example:

16 * 48. Again, this becomes 16 * 16 *3, and any math major worth his salt knows his squares up to 25, just because he's done them so often. So 16 *16 * 3 = 256 * 2 = (250 +6) * 3 = 750 +18 = 768. Again, just three instantaneous steps, if you see them.

Counterexample:

32 * 37. This I would probably just do longform in my head, which I am bad at. No immediately obvious trick (to me at least) so I do the normal three multiplications and one addition.

Example:

Is x<100 prime? Well, I figured out a while ago that you only have to look for factors less than the root of x, so less than 10. This is because if z divides 100, and z is greater than 10, then z*10>100, so z's factor pair (that is, 100/z) must be less than 10. So to find z, you just have to find its pair.

Further, even x's are obvious not prime, and odd numbers are only even the product of two odds, so you only have to look for odd numbers under 10: 3,5,7,9. On top of that, the numbers divisible by 5 and 9 are immediately obvious. So all we have left is to check if a number is divisible by 3 or 7. Well. a number is divisible by 3 if and only if the sum of the digits is, so that is a quick test. That leaves as the only real test divisibility by 7. Thus the question, 'is 83 prime,' can be answered just by noticing that 8 is not divisible by 3, and 83 is 13 more than 70. So pretty much this entire paragraph is already coded into my thought process, which is why the answer will come pretty quickly. The same holds true for most numbers less than 200, you just have to test for 11 and 13 also, and beware of the dreaded 119, which should totally be prime.