Show by induction that if a prime p divides a product of n numbers, then it...

Show by induction that if a prime p divides a product of n numbers, then it divides at least one of the numbers.

Number theory course. Please, I want a clear and neat and readable answer.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Use mathematical induction to prove that If p(x) in F[x] and deg p(x) = n, show...

Use mathematical induction to prove that If p(x) in F[x] and deg p(x) = n, show that the splitting field for p(x) over F has degree at most n!.

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if...

Consider n-dimensional finite affine space: F(n,p) over the field with prime p-elements. a. Show tha if l and l' are two lines in F(n,p) containing the origin 0, then either l intersection l' ={0} or l=l' b. how many points lie on each line through the origin in F(n,p)? c. derive a formula for L(n,p), the number of lines through the origin in F(n,p)

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set...

5. Let n = 60, not a product of distinct prime numbers. Let Bn= the set of all positive divisors of n. Define addition and multiplication to be lcm and gcd as well. Now show that Bn cannot consist of a Boolean algebra under those two operators. Hint: Find the 0 and 1 elements first. Now find an element of Bn whose complement cannot be found to satisfy both equalities, no matter how we define the complement operator.

Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides...

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides n

show that an integer n > 4, is prime iff it is not a divisor of...

show that an integer n > 4, is prime iff it is not a divisor of (n-1)!

a) Prove by induction that if a product of n polynomials is divisible by an irreducible...

a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p(x) then at least one of them is divisible by p(x). You can assume without a proof that this fact is true for two polynomials. b) Give an example of three polynomials a(x), b(x) and c(x), such that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x) nor b(x).

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a...

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.”...

Consider the statement, “For all natural numbers n,n, if nn is prime, then nn is solitary.” You do not need to know what solitary means for this problem, just that it is a property that some numbers have and others do not. Write the converse and the contrapositive of the statement, saying which is which. Note: the original statement claims that an implication is true for all n,n, and it is that implication that we are taking the converse and...

How to show that there are infinitely many prime p of the form p= 1+5k or...

How to show that there are infinitely many prime p of the form p= 1+5k or p=4+5k

Subjects