#3**+1 **

tuffla202206.08.2022

#1**+1 **

On ex. 1 python says:

There are 33 integers in that group, they are:

15, 45, 75, 105, 135, 165, 195, 225, 255, 285, 315, 345,

375, 405, 435, 465, 495, 525, 555, 585, 615, 645, 675,

705, 735, 765, 795, 825, 855, 885, 915, 945, 975

See https://www.online-python.com/nxgpkMajEo

On ex. 2:

Turning to mod,

in python (and other programming languages)

% is used as the modulus operator

ab%20 = 17

(a + 10) x (b + 10)%20 =

(ab +10a + 10b + 100)%20 =

ab%20 + 10a%20 + 10b%20 + 100%20 =

17 + 10 + 10 + 0 = 17 + 20 = 37 (assuming a and b not divisible by 20)

But 37%20 = 17

tuffla202206.08.2022

#1**+1 **

As before, use the onilne calculator, here is the answer:

https://www.emathhelp.net/calculators/algebra-1/polynomial-long-division-calculator/?numer=x%5E10+-+8x%5E8%2B7x%5E4%2B+8x%5E3-12x%5E2+-15x+-+5&denom=+x%5E2+-+1

And an image of the solution:

The fractional expression is the last term in this, so the remainder is -7x -17.

tuffla202206.08.2022

#1**+1 **

You can use an online calculator for that, the solution, and the calculator can be found at:

https://www.emathhelp.net/calculators/algebra-1/polynomial-long-division-calculator/?numer=5x%5E9+-+4x%5E2+%2B2&denom=5x%2B10

Under is screengrab of the answer:

The answer to Your puzzle is thus:

\(-\frac{644x}{5} + \frac{1288}{5} - \frac{2574}{5x + 10}\)

It really is a tedious task to go through all of that.

tuffla202206.08.2022

#2**+1 **

Here is my doodle

https://www.geogebra.org/classic/bhfnxt4x

Heron's formula, and a calculator for it can be found at https://www.mathsisfun.com/geometry/herons-formula.html

tuffla202206.08.2022