Suppose a is a positive integer and p is a prime/ Prove that p|a if and...

Suppose a is a positive integer and p is a prime/ Prove that p|a if and only if the prime factorization of a contains p.

Can someone please show a full proof to this, thank you.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

Please prove 1. Every positive integer is a product of prime numbers. 2. If a and...

Please prove 1. Every positive integer is a product of prime numbers. 2. If a and b are relatively prime, and a|bc, then a|c. 3. The division algorithm for F[x]. Just the existence part only, not the uniqueness part

Let n be a positive integer. Prove that two numbers n2+3n+6 and n2+2n+7 cannot be prime...

Let n be a positive integer. Prove that two numbers n2+3n+6 and n2+2n+7 cannot be prime at the same time.

Let p be an odd prime and a be any integer which is not congruent to...

Let p be an odd prime and a be any integer which is not congruent to 0 modulo p. Prove that the congruence x 2 ≡ −a 2 (mod p) has solutions if and only if p ≡ 1 (mod 4). Hint: Naturally, you may build your proof on the fact that the statement to be proved is valid for the case a = 1.

Suppose that A is a triangle with integer sides and integer area. Prove that the semiperimeter...

Suppose that A is a triangle with integer sides and integer area. Prove that the semiperimeter of A cannot be a prime number. (Hint: Suppose that a natrual number x is a perfect square, and suppose that p is a prime number that divides x. Explain why it must be the case that p divides x an even number of times)

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo...

Let p be an odd prime. (a) (*) Prove that there is a primitive root modulo p2 . (Hint: Use that if a, b have orders n, m, with gcd(n, m) = 1, then ab has order nm.) (b) Prove that for any n, there is a primitive root modulo pn. (c) Explicitly find a primitive root modulo 125. Please do all parts. Thank you in advance

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime...

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime number.

8.Let a and b be integers and d a positive integer. (a) Prove that if d...

8.Let a and b be integers and d a positive integer. (a) Prove that if d divides a and d divides b, then d divides both a + b and a − b. (b) Is the converse of the above true? If so, prove it. If not, give a specific example of a, b, d showing that the converse is false. 9. Let a, b, c, m, n be integers. Prove that if a divides each of b and c,...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of...

Let t be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having v varieties, then there exists a Steiner triple system having v^t varieties

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod...

Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod 4) or n ≡ 3 (mod 4), then n is not a perfect square.

Subjects