With the Euclidean algorithm we finally have an efficient algorithm for finding the multiplicative inverse in Zm that is much better than exhaustive search. Find the inverses in Zm of the following elements a modulo m: 1. a=7, m=26 (affine cipher) 2. a=19, m=999

a) The inverse of 7 in is 15

b) The inverse of 19 in is 631

Step-by-step explanation:

1)

26-3*7 = 5

7-5 = 2

5 - 2*2 = 1

Thus 1 = 5 - (2*(7-5) ) = 3 * 5 - 2*7 = 3 * (26-3*7) - 2*7 = 3*26 -11*7

Now, lets check 7 * (-11) = -77, and -63 + 3*26 = -77+78 = 1. We take as inverse -11+26 = 15 (so that it lies between 1 and 26).

2)

999/19 = 52.278....

999 - 52*19 = 11

19 - 11 = 8

11 - 8=3

8 - 2*3 = 2

3 - 2 = 1

Thus,

1 = 3-2 = 3 - (8-2*3) = 3*3-8 = 3*(11-8) - 8 = 3*11 - 4*8 = 3*11 - 4*(19-11) = 7*11-4*19 = 7*(999-52*19) - 4*19 = 7*999 - 368 * 19  

We take as inverse -368+999 = 631.

Lets check, -368*19 + 7 *999 = -6992 + 6993 = 1, it works.

no oblique asymptote

step-by-step explanation:

to find the oblique asymptote of f(x)

here numerator has degree 1 and denomintor degree 2

degree of numerator < degree of denominator

hence there is no oblique asymptote

when the degree of the denominator > that of the numerator, we saw that the value in the denominator got so much bigger, so quickly,   giving us a horizontal asymptote of the x-axis. or we can write

f(x) = a polynomial of x + a proper fraction.

in this case only we have oblique asymptotes but not when numerator has less degree.