Answer:
The prove is as given below
Step-by-step explanation:
Suppose there are only finitely many primes of the form 4k + 3, say {p1, . . . , pk}. Let P denote their product. 
Suppose k is even. Then P ≅ 3^k (mod 4) = 9^k/2 (mod 4) = 1 (mod 4). 
ThenP + 2 ≅3 (mod 4), has to have a prime factor of the form 4k + 3. But pₓ≠P + 2 for all 1 ≤ i ≤ k as pₓ| P and pₓ≠2. This is a contradiction. 
Suppose k is odd. Then P ≅ 3^k (mod 4) = 9^k/2 (mod 4) = 1 (mod 4). 
Then P + 4 ≅3 (mod 4), has to have a prime factor of the form 4k + 3. But pₓ≠P + 4 for all 1 ≤ i ≤ k as pₓ| P and pₓ≠4. This is a contradiction. 
So this indicates that there are infinite prime numbers of the form 4k+3.
 
        
                    
             
        
        
        
I'll help with number 5 since number 4 is a bit too small and squished to read.
Check out the attached image below for the full two-column proof. The added entries are in red.
The given statement is basically what the teacher shows at the top of the problem, which is the fact that angle 5 and angle 2 are congruent. You just repeat this statement.  
The reason for statement 2 is that angle 2 and angle 4 are vertical angles. They are opposite angles formed by a pair of intersecting lines. This is the vertical angle theorem at work. We'll use the vertical angle theorem again for statement 4 when we say that angle 5 and angle 8 are the same measure.
The transitive property comes into play when we connect angles 2, 5, and 4 together, which helps us get line 3. We also connect angles 4, 5 and 8 together to get the last line, which is why the reason for statement 5 is "transitive property"
 
        
             
        
        
        
There are 72 passengers in total and 35 are men and women as one category. 
The number of children are 72 - 35 = 37 

So the success is 37 (number of children. And the total number of people is 72. 
The probability of a child being asked is 37/72.
 
        
             
        
        
        
Answer: y−3=−3y−43
Step 1: Add 3y to both sides.
y−3+3y=−3y−43+3y
4y−3=−43
Step 2: Add 3 to both sides.
4y−3+3=−43+3
4y=−40
Step 3: Divide both sides by 4.
4y
4
=
−40
4
y=−10
Answer:
y=−10
Step-by-step explanation: brainliest:)
 
        
             
        
        
        
I'm assuming there's a chart, but if there isn't,
The population will continue to increase and decrease, as the population will grow until it's reached its carrying capacity, and then decrease because there aren't enough resources, and then increase, then decrease, etc.