# 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. It can be used in full or in parts.

In estimating products or quotients, we often use only “the first by the first” while we use “the last by the last” when in adding whole numbers in a column, we align the numbers by their last digits. Some divisibility tests require that we look at only the last 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**.

In combination with another sub-Sutra, ** the Product of the Sums is the Sum of the Product**,

**FFLL**becomes a powerful tool for checking the results of arithmetic and algebraic calculations, factoring, and determining square roots of perfect square polynomials.

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 much nearer to **49**, the square of **7** than to **64**, the square of **8**. The square root must be **71**.

Our illustration shows how we can quickly determine the other factor by just applying ** the First by the First and the Last by the Last**.

A quick look at the given polynomials will tell us that the other factor is a binomial. Dividing the first term of the quadrinomial by the first term of the trinomial, **6x ^{3} ÷ 3x^{2}** will give the first term of the binomial factor,

**2x**while dividing the last terms

**-15 ÷ 5**will give the last term of the binomial,

**-3**. So (2x – 3) is the other factor.