6.6. Using the extended Euclidean algorithm, compute the greatest common divisor and the parameters s, t of 1. 198 and 243 2. 1819 and 3587 For every problem check if s r0 +t r1 = gcd(r0,r1) is actually fulfilled. The rules are the same as above: use a pocket calculator and show what happens in every iteration step.

1) The gcd is 9

9 = 9*243-11*198

2) The gcd is 17

17 = -36*3587+71*1819

Step-by-step explanation:

1) First, we divide the greater number with the smallest:

243/198 = 1.227

The whole value is 1, and the remainder is243-1*198 = 45

Now we take the second number, 198 and 45 and repeat the same process:

198/45 = 4.4

The whole part is 4

the remainder is

198 - 4*45 = 18

Now we take 45 and 18 and repeat the process

45/18 = 2.5

the whole part is 2, the remainder is 45-2*18 = 9

Now,

18/9 = 2 is a whole number, so we stop in 9, which is the gcd.

How we find s,t?

We have to start from 9 and go upwards, replacing in bigger numbers (used in our algorithm) if possible

9 = 45-2*18

Now, we replace 18, using 18 = 198-4*45

9 = 45-2*18 = 45- 2*(198-4*45) = 9*45 - 2*198

Now, time to replace 45, using 45 = 243-198

9 = 9*45-2*198 = 9*(243-198) - 2*198 = 9*243 - 11*198

If we make the computation 9*243-11*198 = 2187-2178 = 9

Thus, the gcd is 9 and the parameters are 9 for 243 and -11 for 198.

2)

Lets divide 3587 with 1819

3587/1819 = 1.971...

The whole part is 1, the remainder is

3587-1*1819 = 1768

Now, its the turn for 1819 and 1768

clearly, the whole part oof the division will be 1, since ther are similar numbers. The remainder is 1819-1768 = 51

Now, lets continue with 1768 and 51

1768/51 = 34.66666

The whole part is 34, the remainder is

1768-34*51 = 34

Now, between 51 and 34:

The whole part of 51/34 is 1 and the remainder is 51-34 = 17

And between 34 and 17:

34/17 = 2, which is a whole number. So we stop in 17.

The gcd is 17, lets find now the parameters. We start by replacing 17

17 = 51 - 1*34

Now 34: 34 = 1768-34*51

51-34 = 51-(1768-34*51) = 35*51-1768

Now, we replace 51 = 1819-1768:

35*51-1768 = 35*(1819-1768) - 1768 = 35*1819-36*1768

And last, we replace 1768 with 3587-1819

35*1819-36*1768 = 35*1819-36*(3587-1819) = 71*1819 - 36*3587

So, the parameters for 3587 and 1819 are -36 and 71 respectively and the gcd is 17.

Lets check:

-36*3587+71*1819 = -129132+129149 = 17

step-by-step explanation:

the distance (d) between two points (x1,y1) and (x2,y2) is given by the formula

d = √   ((x2-x1)2+(y2-y1)2)

d = √ (-11-6)2+(4-3)2

d = √ )2+(1)2)

d = √ (289+1)

d = √ 290

the distance between the points is 17.0293863659264

the midpoint of two points is given by the formula

midpoint= ((x1+x2)/2,(y1+y2)/2)

find the x value of the midpoint

xm=(x1+x2)/2

xm=(6+-11)/2=-2.5

find the y value of the midpoint

ym=(y1+y2)/2

ym=(3+4)/2=3.5

the midpoint is: (-2.5,3.5)

graphing the two points, midpoint and distance

p1 (6,3)

p2 (-11,4)

midpoint (-2.5,3.5)

the length of the black line is the distance between the points (17.0293863659264)

find the slope of the line connecting the two points

slope =   (y2-y1)   =   (4-3)   =   (1)   = -0.0588235294117647

(x2-x1) (-11-6) (-17)

find the equation of the line passing through the two points

the general equation for a straight line is

y = mx + b

where m represents the slope of the line which we found in the previous step to be -0.0588235294117647

y = -0.06x + b

we substitute x and y for the values from one of our points (6,3)

3 = -.06×6 + b

3 = -0.35 + b

3--0.35 = b

3.35 = b

knowing both b and m, we can contruct the equation of the line

y= -0.06x+ 3.35

x and y intercepts

the x-intercept is a point on the graph where y is zero

using the equation we found in the previous step and substituting zero for y

y= -0.06x+ 3.35

0= -0.06x+ 3.35

0.06x= 3.35

x= 3.35/0.06 = 57.00

the x intercept for this straight line is 57.00

the y-intercept is a point on the graph where x is zero

using the equation we found in the previous step and substituting zero for x

y= -0.06×0+ 3.35

y= 3.35

the y intercept for this straight line is 57.00

the distance (d) between two points (x1,y1) and (x2,y2) is given by the formula

d = √   ((x2-x1)2+(y2-y1)2)

d = √ (-11-6)2+(4-3)2

d = √ )2+(1)2)

d = √ (289+1)

d = √ 290

the distance between the points is 17.0293863659264

the midpoint of two points is given by the formula

midpoint= ((x1+x2)/2,(y1+y2)/2)

find the x value of the midpoint

xm=(x1+x2)/2

xm=(6+-11)/2=-2.5

find the y value of the midpoint

ym=(y1+y2)/2

ym=(3+4)/2=3.5

the midpoint is: (-2.5,3.5)

graphing the two points, midpoint and distance

p1 (6,3)

p2 (-11,4)

midpoint (-2.5,3.5)

the length of the black line is the distance between the points (17.0293863659264)

find the slope of the line connecting the two points

slope =   (y2-y1)   =   (4-3)   =   (1)   = -0.0588235294117647

(x2-x1) (-11-6) (-17)

find the equation of the line passing through the two points

the general equation for a straight line is

y = mx + b

where m represents the slope of the line which we found in the previous step to be -0.0588235294117647

y = -0.06x + b

we substitute x and y for the values from one of our points (6,3)

3 = -.06×6 + b

3 = -0.35 + b

3--0.35 = b

3.35 = b

knowing both b and m, we can contruct the equation of the line

y= -0.06x+ 3.35

x and y intercepts

the x-intercept is a point on the graph where y is zero

using the equation we found in the previous step and substituting zero for y

y= -0.06x+ 3.35

0= -0.06x+ 3.35

0.06x= 3.35

x= 3.35/0.06 = 57.00

the x intercept for this straight line is 57.00

the y-intercept is a point on the graph where x is zero

using the equation we found in the previous step and substituting zero for x

y= -0.06×0+ 3.35

y= 3.35

the y intercept for this straight line is 57.00 and therefore the awnser is c.