# MSC #17- By Alternate Elimination and Retention

For the 17^{th} MATH-Inic Special for Christmas, we are introducing the Vedic Math sub-Sutra, “By Alternate Elimination and Retention”. In order to simplify solution of some problems, we sometimes ignore or temporarily eliminate some variables and retain only the others. One of the most common applications of this sutra is in finding the intercepts of curves on an axis by equating a variable to zero.

In combination with other Sutras like “By Addition and By subtraction” and “Proportionately”, we can use it to find the Highest Common Factor (or Greatest Common Divisor) of numbers and polynomials by addition or subtraction of multiples of the numbers. This method is similar but more flexible than the Euclidian algorithm used to solve this type of problem.

For instance, if we want to find the GCD of 589 and 437, we can first find their difference: 589 – 437 = 152. Now we can do one of two things: 1) we can subtract twice 152 from 437, 437 – 2 x 152 = 437 – 304 = 133 and then perform this subtraction: 152 – 133 = 19 or 2) subtract 437 from 3 times 152: 3 x 152 – 437 = 456 – 437 = 19. In both cases we obtained 19 which we can verify as the HCF of the two numbers.

We can also create zeroes at the end of the numbers if we add 3 times 437 to 589, we will get 3 x 437 + 589 = 1311 + 589 = 1900. Obviously 100 is not a factor of the 2 numbers but 19 can be.

In our featured example we alternately eliminated the highest and lowest terms of the polynomials to get their Highest Common Factor, as shown in our infographics.

More applications are discussed in Chapter 17 of our forthcoming book “30 Master Strategies in Computing”.