Paraavartya Yojayet:

Paraavartya Yojayet is the fourth sutra of Vedic mathematics. Its corollary is Kevalaih Saptakam Gunyat.

This method is related to the Chinese remainder theorem and the Horner's rule of synthetic division, but arguably has even more applications.

What follows is a brief and incomplete summary of the math shortcuts this sutra contains:

Divisor is more than one digit and slightly higher than a power of 10
Example: 12345 divided by 12
Subtract the divisor (12) from the nearest power of ten: 10 - 12 = -2
We will need to keep -2 in mind for the next steps.
Separate the last digit (5) from the preceeding digits (1234). The last digit will be used later to calculate the remainder, while the preceeding digits will be used to calculate the quotient.
The first digit of the dividend will be the same (in most cases) as the first digit of the quotient, so for now we will assume it is. So the quotient we have so far is 1xxx
Multiply this first digit of the quotient (1) by -2. 1 x -2 = -2
Add that number to the second digit of 1234: -2 + 2 = 0 So the quotient we have now is 10xx
Multiply this second digit of the quotient (0) by -2. That equals 0. Add that number to the third digit of 1234. That equals 3
So the quotient we have now is 103x
Multiply this third digit of the quotient (3) by -2. This equals -6. Add this number to the fourth digit of 1234. -6 + 4 = -2
Since this is a negative number, we will write the quotient as 1030 and then add this negative number: 1030 + -2 = 1028. This is the quotient.
Now let's return to the last digit of the dividend that we set aside: 5
Multiply the last digit of the quotient by -2. Important Note: always use the first number we arrived at before adding the negative number to 1030: -2
-2 x -2 = 4
4 + 5 = 9
Thus, the quotient is 1028 and the remainder is 9.

This method can also be used for numbers slightly higher than 100, 1000, etc.
For a more detailed explaination of Paravartya - Yojayet, see:
http://www.vedamu.org/Mathematics/MathematicalFormulae/Sutras/Paravartyayogayet.asp

Applications in algebra
This sutra can be used to simplify algebraic equations. For information on that method visit the link above or the resources below.

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm