Question

In: Advanced Math

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c. (B) Let p ≥ 2....

(A) Let a,b,c∈Z. Prove that if gcd(a,b)=1 and a∣bc, then a∣c.

(B) Let p ≥ 2. Prove that if 2^p−1 is prime, then p must also be prime.

(Abstract Algebra)

Solutions

Expert Solution

(A)

Let and suppose that and .

and

multiply above equation by c ,

now

(B)

Let is prime .If possible suppose that p is not prime that is p is composite number then ther exists r and s such that

now

put

it is clear that

since .

which shows that ,then is the product of terms greater than 1 .

Hence is composite number which is a contradiction that is prime number .

Our supposition is wrong ,hence p must be prime .

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Prove that gcd(a,b) = gcd(a+b,lcm(a,b))

Bezout’s Theorem and the Fundamental Theorem of Arithmetic 1. Let a, b, c ∈ Z. Prove...

Bezout’s Theorem and the Fundamental Theorem of Arithmetic 1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if and only if gcd(a, b)|c. 2. Prove that if c|ab and gcd(a, c) = 1, then c|b. 3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if and only if a and b have no prime factors in common.

Let A and B be events with P(A) = 0.5, P(Bc ) = 0.4, P(Ac ∩...

Let A and B be events with P(A) = 0.5, P(Bc ) = 0.4, P(Ac ∩ Bc ) = 0.3. (a) Calculate P(A ∪ B), P(A ∩ B), P(B|(A ∪ B)), and P(Ac |B). b) Are A and B independent? Explain why.

Let A = Z and let a, b ∈ A. Prove if the following binary operations...

Let A = Z and let a, b ∈ A. Prove if the following binary operations are (i) commutative, (2) if they are associative and (3) if they have an identity (if the operations has an identity, give the identity or show that the operation has no identity). (a) (3 points) f(a, b) = a + b + 1 (b) (3 points) f(a, b) = a(b + 1) (c) (3 points) f(a, b) = x2 + xy + y2

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a)...

Let Z[ √ 2] = {a + b √ 2 | a, b ∈ Z}. (a) Prove that Z[ √ 2] is a subring of R. (b) Find a unit in Z[ √ 2] that is different than 1 or −1.

Q. Let A, B independent events, with P(A) = 1/2 and P(B) = 2/3. Now C...