Answer:
Check the explanation
Step-by-step explanation:
(a)Let p be the smallest prime divisor of (n!)^2+1 if p<=n then p|n! Hence p can not divide (n!)^2+1. Hence p>n
(b) (n!)^2=-1 mod p now by format theorem (n!)^(p-1)= 1 mod p ( as p doesn't divide (n!)^2)
Hence (-1)^(p-1)/2= 1 mod p hence [ as p-1/2 is an integer] and hence( p-1)/2 is even number hence p is of the form 4k+1
(C) now let p be the largest prime of the form 4k+1 consider x= (p!)^2+1 . Let q be the smallest prime dividing x . By the previous exercises q> p and q is also of the form 4k+1 hence contradiction. Hence P_1 is infinite
Answer:
-4
Step-by-step explanation:
Let the width be w, then length = 2w - 1
Area = length times width = w(2w - 1) = 21.
Thus, 2w^2 - w = 21 or 2w^2 - w - 21 = 0
Solving the quadratic equation gives that w = 3.5.
Therefore, width is 3.5 ft and length is 2(3.5) - 1 = 6 ft.
Answer:
Step-by-step explanation:
1) Let the random time variable, X = 45min; mean, ∪ = 30min; standard deviation, α = 15min
By comparing P(0 ≤ Z ≤ 30)
P(Z ≤ X - ∪/α) = P(Z ≤ 45 - 30/15) = P( Z ≤ 1)
Using Table
P(0 ≤ Z ≤ 1) = 0.3413
P(Z > 1) = (0.5 - 0.3413) = 0.1537
∴ P(Z > 45) = 0.1537
2) By compering (0 ≤ Z ≤ 15) ( that is 4:15pm)
P(Z ≤ 15 - 30/15) = P(Z ≤ -1)
Using Table
P(-1 ≤ Z ≤ 0) = 0.3413
P(Z < 1) = (0.5 - 0.3413) = 0.1587
∴ P(Z < 15) = 0.1587
3) By comparing P(0 ≤ Z ≤ 60) (that is for 5:00pm)
P(Z ≤ 60 - 30/15) = P(Z ≤ 2)
Using Table
P(0 ≤ Z ≤ 1) = 0.4772
P(Z > 1) = (0.5 - 0.4772) = 0.0228
∴ P(Z > 60) = 0.0228
"Per" essentially means "divided by." To find pieces per ounce, divide pieces by ounces.
(252 pieces)/(14 ounces) = (252/14) pieces/ounce = 18 pieces/ounce
