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)

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 2 years ago

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