Question

In: Advanced Math

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)!

Expert Solution

Suppose be an such that

Suppose is prime .

divides a number which are only mutiple of i.e.,

All of does divisible by .

does not divisible .

does not divisible by .

Hence is not a divisor of .

Conversely suppose is not a divisor of .

If is not prime then for some with

divides and also divides as

divides .

Hence is prime if and only if it is not a divisor of .

Colby Messinger answered 3 years ago

A positive integer n is said to be prime (or, "a prime") if and only if...

A positive integer n is said to be prime (or, "a prime") if and only if n is greater than 1 and is divisible only by 1 and n . For example, the integers 17 and 29 are prime, but 4, 21 and 38 are not prime. Write a function named "is_prime" that takes a positive integer argument and returns as its value the integer 1 if the argument is prime and returns the integer 0 otherwise. Can you make...

Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

ATestforPrimalityisthefollowing: Given an integer n > 1, to test whether n is prime check to see...

ATestforPrimalityisthefollowing: Given an integer n > 1, to test whether n is prime check to see if it is divisible by a prime number less than or equal to it’s square root. If it is not divisible by an of these numbers then it is prime. We will show that this is a valid test. prove ∀n, r, s ∈ N+, r s ≤ n → (r ≤ √n ∨ s ≤ √n) b) Prove ∀ n ∈ N+ ,...

For all integers n > 2, show that the number of integer partitions of n in...

For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

(a) Let a > 1 be an integer. Prove that any composite divisor of a −...

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a. (b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a. (c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be...

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.

4. Let n ≥ 8 be an even integer and let k be an integer with...

4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤ k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . , n}. How many such subsets contain at least two even numbers?

Show that for any square-free integer n > 1, √ n is an irrational number

(Prime Numbers) An integer is said to be prime if it is divisible by only 1...

(Prime Numbers) An integer is said to be prime if it is divisible by only 1 and itself. For example, 2, 3, 5 and 7 are prime, but 4, 6, 8 and 9 are not. Write pseudocode and function called isPrime that receives an integer and determines whether the integer is prime or not. Write a test program that uses isPrime to determine and print all the prime numbers between 1 and 1000. Display 10 numbers per line. Twin primes...

(a) Let n = 2k be an even integer. Show that x = rk is an...

(a) Let n = 2k be an even integer. Show that x = rk is an element of order 2 which commutes with every element of Dn. (b) Let n = 2k be an even integer. Show that x = rk is the unique non-identity element which commutes with every element of Dn. (c) If n is an odd integer, show that the identity is the only element of Dn which commutes with every element of Dn.