Casting Out Nines

How do you check the answer of a large multiplication like 8216*4215=36430440 without actually doing the multiplication.

Sum 8+2+1+6=17. Take modulo 9 = 8.
Sum 4+2+1+5=12. Take modulo 9 = 3.

Sum 3+6+4+3+4+4=24. Take modulo 9 = 6.

Now check 8*3=24 and 2+4 = 6.

Austrian Method of Subtraction

How do you subtract 5592 from 11149. Simple question....

There's an interesting method which involves only addition as it is easier to do than subtraction. Add from the amount to be subtracted upto the next higher unit, then to the next and so on.

Example 5592 + 8 = 5600. 5600 + 400 = 6000. 6000 + 5000 = 11000. 11000 + 149 = 11149.
so the answer is 8+400+5000+149 = 5557.

Pascal's Triangle

There's a very good article about Pascal's Triangle in Wiki. I am just enumerating some important points for quick reference.

- Construction of Triangle, value of the element is equal to the sum of the above two adjecent values.

- Sum of the elements of the mth row is equal to 2 raised to power m-1.

- Elements of mth row correspond to coefficients of 11 raised to power m-1.

- Generally the elements of mth row are the coefficients of the binomial expression (x+y) raised to power m-1.

- The diagonals next to the edge diagonals contain the natural numbers in order. Moving inwards, the next pair of diagonals contain the triangular numbers in order.

- Starting at the top 1, without backtracking or going sideways, try to get to another node via these grid paths as many ways as possible. The answer is whatever number the node has.

- Sum of a shallow diagonal is a Fibonacci's number.

- In rows where the second number (the 1st number following 1) is prime, all the terms in that row except the 1s are multiples of that prime.

- Pascal's triangle can be used as a lookup table for the number of arbitrarily dimensioned elements within a single arbitrarily dimensioned version of a triangle (known as a simplex).

- No one knows whether any number occurs 5 times in a Pascal's triangle.