Applications in Number Theory

Greatest common divisor as a linear combination Theorem If a and b are positive integers and gcd(a, b) = d then there are integers s and t such that d = s×a + t×b. We illustrate first a method of finding these multipliers s and t by reversing the calculations of the Euclidean Algorithm. Later we show a direct way of finding s and t using the Extended Euclidean Algorithm. The Extended Euclidean Algorithm can be used to find the gcd of two numbers and express it as a linear combination of those numbers. It uses auxiliary numbers 1 and 0 and two starting conditions to produce an invariant expression G = S×A + T×B that yields the desired result. Example We can show that gcd(356, 252) = 4 and that 4 = (17)356 + (-24)252 In the following tableau, the first two lines express A and B as linear combinations of themselves. The calculation begins in the third line where Q n = floor (R (n–2) / R (n–1)) and A = R (-1) and B = R (0). Each of the other columns uses Q n to find the subsequent entry, and the process is repeated for each line. A 356 1 0 356 = 1HA + 0HB B 252 0 1 252 = 0HA + 1HB