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.

Expert Solution

Colby Messinger answered 2 years ago

Let G be a finite group and p be a prime number.Suppose that pr divides the...

Let G be a finite group and p be a prime number.Suppose that pr divides the order of G. Show that G has a proper subgroup of order pr.

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

Let gcd(a, p) = 1 with p a prime. Show that if a has at least...

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

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)

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