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.
Answer:
4 square feet
Step-by-step explanation:
We can see that the dimensions of the hole in the carpet are 2ft by 2ft. All we have to do is find the area of that shape. Since the two side lengths are the same, and the shape appears to have 4 right angles in each of its corners, we can deduce that the shape is a square. We know that the area of a square can be found by multiplying the length and width measurements of the shape:
2ft * 2ft
= 4 ft²
your answer would be c. y times parentheses 5 plus y times a. C. y • (5 + y) • a
I hope this helps
Part A) x-intercepts simply show that when the value of the function is zero. Vertex coordinates show that when the function obtains its maximum value. When x=50, function obtains its maximum value and it's 75. The function is increasing in the interval (0, 50) and decreasing in the interval (50, 100). In regard to the height and distance of the tunnel, these numbers show that decreasing and increasing intervals are symmetric. Each number from the intervals has its own pair in the corresponding interval and they are located in the same distance from the midpoint (50,75)
Part B) In order to calculate the average rate of change, we can first write the function. Using the information about the x-intercept and the vertex coordinates, we find that our function is

.
Plugging 15 and 35 in x, we can find the values of the function, i.e.

and

.
Then, the average change is