QUESTION 1.
1A. Prove that GCD(a, a+2) = 1 if a is odd, and GCD(a, a+2) = 2 if a is even

1B. Show that if a≡b(mod n) and if b≡c(mod n), then a≡c(mod n)

1C. Simplify the following congruences:
15x≡9 (mod 25)
6x≡3 (mod 9)
14x≡42 (mod 50)

1D. Describe the general solution for x and y, if it exists: 35x+47y=1

Request clarification:

Expert:

We answer only one question at a time. You decide 

User:

Hello, this is one question with 4 parts. If not possible to answer all please answer part C and D

Expert:

I can solve Only D.. do you want it

User:

Okay then, is there another part you can also solve? If yes, please assist also

Cost of 7 gallons=$24.50

Cost of 1 gallon=24.50/7=3.5

Cost of 15 gallons=15*3.5=52.5

Cost of 15 gallons will be $52.5

The answer is in the image 

The answer is in the image 

F=ma

where F=force

m=mass

a=acceleration

Here,

F=4300

a=3.3m/s2

m=F/a

    =4300/3.3

    =1303.03kg

The solution is given in the image below

Salesperson will make 6% of 1800

=(6/100)*1800

=108

Salesperson will make $108 in $1800 sales

Given height of gymnasium is 5/6 height of 30 foot ball

therefore height of gymnasium=5/6 * 30

=25 feet

We can put the numbers in ascending order-

36, 38, 39, 39, 40, 47, 48, 51

Range=max-min

          =51-36

          =15

There are 6 outcomes 1,2,3,4,5,6

Total outcomes = 6

Outcomes when rolling a number is 4 or less = 1,2,3,4

Total outcome when rolling a number is 4 or less=4

Probability = required outcome/total outcome

                 =4/6

                 =2/3

Therefore probability = 2/3

(​1 or 7) ; number of possible outcome = 2

(a, e,​ i, o, or​ u) ; number of possible outcome = 5

So,

Number of possible outcomes = 2 x 5

Number of possible outcome =10