# MSC #18 – The First by the First and the Last by the Last

The Vedic Math sub-Sutra, “First by the First and the Last by the Last”(FFLL) has varied applications in Mathematics. When adding whole numbers in a column, we align the numbers by their last digits. Many divisibility tests require that we look at only the last digits. In estimating products and quotients, we often consider only the first digits. Certain multiplication short cuts are developed if the first and last digits follow certain rules. Two-digit square roots and cube roots of perfect squares and cubes can be easily identified using FFLL.

For example, if it is given that 5,041 is a perfect square, its square root can be easily determined by partitioning the square in groups of two digits starting from the right. The last digit of the square is 1, so the last digit of the square root must be either 1 or 9. The first two digits, 50, is just a little bigger than 49, the square of 7. The square root must be 71.

In combination with another sub-Sutra, “the Product of the Sums, is the Sum of the Product”, FFLL becomes a powerful tool for checking, factoring and determining square roots of perfect square polynomials.

In our example, we quickly determined the other factor by just applying “the First by the First and the Last by the Last”

More examples of the application of this Sutra is given in Chapter 18 of our forthcoming book, “30 Master Strategies in Computing”