2.3. Let g be a primitive root for Fp. (a) Suppose that x = a and x = b are both integer solutions to the congruence gx ≡ h (mod p). Prove that a ≡ b (mod p − 1). Explain why this implies thatHoffstein, Jeffrey.

Let p be a prime number. Suppose that a and n are positive numbers such that gcd(a, p) = 1 and gcd(n, p − 1) = 1.

Determine the number of solutions of the congruence x n ≡ a (mod p). Solution.

Let g be a primitive root modulo p. Since gcd(a, p) = 1, we can find an integer l such that a ≡ g l (mod p). In addition, since x ≡ 0 (mod p) is not a solution, we can write x ≡ g k (mod p) for some integer k.

Now the congruence can be written as g kn ≡ g l (mod p), which is equivalent to kn ≡ l (mod p − 1). This congruence can be solved in k by k ≡ ln−1 (mod p − 1) as gcd(n, p − 1) = 1, yielding a unique solution x ≡ g k (mod p) of the congruence x n ≡ a (mod p).

Step-by-step explanation:

d. $180

step-by-step explanation:

if the dimensions of the room is 9ft by 12 ft, we have the area of the floor as,

so the area of the floor is 108 square feet, converting that into square yards we get,

the cost of carpet is $15 per square yard, therefore the cost of carpet for 12 square yards is,

hence, the cost of carpet is $180.

assuming the dimensions of the room be,

15 ft by 12 ft

so, the area of the floor of that room is,

we can convert 180 square feet into square yards, we know that,

1 sq ft = 0.111 sq yard

so, 180 square feet is,

so the area of floor is 20 square yards.

now, the cost of carpet is 15 dollars per square yard, therefore the cost of carpet for 20 square yards is,

so, it will cost $300.

answer: i dont know but if you try hard you will get it right : )

step-by-step explanation: