Prove or Disprove: Let p be a prime and a an integer. If p^8 | a^5, then p^2 | a.

Q: Prove using mathematical induction that (n + 1)! > 2"-1, for any integer n > 1.

A:

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: Show that there are infinitely many primes.

A:

Q: are there any even numbers greater than 2 which are prime? why or why not?

A: We will find the solution in the following step

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: Establish by mathematical induction that any postage of at least 12c can be obtained using 3c and 7c…

A: We have to establish by mathematical induction that any postage of at least 12c can be obtainedusing…

Q: Pigeonhole Principle Prove that among any 100 integers there always two whose difference is…

A: The given problem is to prove the following statement which follows the pigeonhole principle, we…

Q: Prove by induction, '2 divides n2+n whenever n is a positive intege

A:

Q: Prove that for every integer n, if n^2 is divisible by 2, then n is divisible by 2.  Make sure that…

A:

Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: Prove or disprove: There are inline number of primes

A: Known fact:A prime is a natural number whose factors are 1 and itself.

Q: Prove by the principle of mathematical induction for all natural number 'n': 1 + 4 + 7 + 1 - - - - +…

A: We need to use Principle of mathematical induction to prove  1+4+7+...+(3n-2)=12n3n-1      ...(1) We…

Q: Let p be a prime larger than 3. 1. Show that the remainder, when p is divided by 6, is 1 or 5. Note:…

A:

Q: Use induction to prove that 2*3^(n-1) is less than or equal to 2n choose n (picture attached)

A: We have to prove 2⋅3n−1≤2nn by using of principal of mathematical induction, where n≥0 Recall that…

Q: primes

A:

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Prove that there are no 3 consecutive odd prime numbers, that is, for any odd p, at least one of…

A: GIVEN: Prove that there are no 3 consecutive odd prime numbers

Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

A:

Q: Find (characterize) all primes p > 3 that satisfy (금) 1.

A:

Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

A:

Q: Only one prime of the form1+4n exists. Determine this prime number and prove it's the only one of…

A: A number that is greater than 1 is said to be prime if it has only two factors 1 and the number…

Q: fermat's theorm states, if p is prime & a is a positive integer not divisible by p then-------- and…

A: Fermat's Theorem: If p is a prime and a is a positive integer and not divisible by the prime p, then…

Q: Show that n-lis divisible by 8, if n is an odd positive integer

A:

Q: 5. Give a recursive definition of the set of positive integers that are multiples of 5.

A: Solution

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

A:

Q: 24. Show that if p is a prime and k is an integer such that Isksp-1.then p divides ().

A: See the handwritten solution

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

A:

Q: Describe a general method for finding all the divisors of a positive integer once you know the prime…

A: Given  A positive Integer whose prime factorization is given . To find all the divisors with general…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: 7. What is the second step In proving by mathematical induction that for every positive Integer n,…

A:

Q: Use permutations to solve. From a cooler with 9 different kinds of soda, 3 are selected for 3…

A: The formula of permutation is: Prn=n!n-r! This is the required answer.

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: 4. A property that, two consecutive positive integers are relatively prime. Consider any two…

A: Two consecutive numbers are said to be relatively prime or coprime numbers if they have no common…

Q: ition to prove the n positive number. of Your Your answe that 1/x 00 as

A:

Q: Let n > 1 be an integer. Show that if pn for all primes p≤ √n, then n is either prime or the product…

A: Introduction: Proof by contradiction, sometimes referred to as indirect proving or the reductio ad…

Q: yields a true statement when n is replaced by any positive integer.

A: The above statement is true. Step-2 i explained in detailed explanation

Q: Fermat's theorem is true for only prime number. إختر واحداً.

A:

Q: 2- Using Direct proofsprove if odd intege mis an even integer then mn is also an even integer

A:

Q: Use mathematical induction to prove 8^n − 1 is a multiple of 7 whenever n is a positive integer.

A:

Q: How many natural number that are less or equal to 1000 and that are not multiplea of 3,5or7 exist

A: To find: Number of natural numbers ≤1000 that are not multiple of 3 or 5 or 7.

Q: 5. Use ordinary induction to prove that for every positive integer n, n°-n is a multiple of 6. Only…

A: To prove that  For every positive integer n, n^3-n is a multiple of 6 by using induction method.

Q: Fermat's Little Theorem tells us that if p is prime and a is an integer not divisible by p, then…

A:

Q: Fermat's theorem is not true for some prime number. اختر واحد

A: False