Use the test for primality to determine whether the following numbers are prime or not. Test for Primality 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 its square root. If it is not divisible by any of these numbers, then it is prime. Write each number as a product of prime numbers. If the number is prime, enter PRIME. 493=   733=   209=   419=

Q: Use the test for primality to determine whether the following numbers are prime or not. Test for…

A: Solving for 319.  Primality check319 = 17.86Hence prime number less than 17.86…

Q: a) Give an example of a number m possessing exactly 12 primitive roots. b) How many primitive roots…

A: As per company guidelines I have solved first three sub parts if you want to get answer of the 4th…

Q: Prove that for all natural number n, 52n-1 is divisible by 8.

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Q: Use mathematical induction to prove that 3n²+1 is divisible by 4 whenever n is an odd number.

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Q: prove the following statement: Let n be an integer. If a and b are integers such that a is divisible…

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Q: If a and b are two distinct prime numbers, then the greatest common divisor of a and b is O ab

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Q: Let p > 5 be the prime number. Prove that the expression p – 10p +9 is divisible by 1920.

A: We have to prove that the expression p4-10p2+9 is divisible by 1920.

Q: Use the Mathematical induction to verify that , for n any positive integer, 6" – 1 is divisible by 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Consider a product of an odd prime with itself and represent it as a sum of two squares. 1. 13^2 =…

A: According to question given:Product of an odd prime with itself and represent it as a sum of Two…

Q: 4. Prove that every odd natural number is of one of the forms 4n+ 1 or 4n+3, where n is an integer.…

A: Disclaimer:  Since you have asked multiple question, we will solve the first question (Number 4) for…

Q: Find the canonical prime factorization and solve for No. of positive divisor and sum of positive of…

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Q: Every two different prime numbers are relatively prime. False True

A: In this question we need to check the given statement is true or false

Q: Which of the following correctly describes the Fundamental Theorem of Arithmetic? Every positive…

A: Here we have to find which is correct statement of fundamental theorem of arithmetic.

Q: 2. Give an example to show that the following conjecture is not true: Every positive integer can be…

A: Every positive integer can be written in the form p+a2, where p is either a prime or 1, and a≥0 We…

Q: Use the following statement to answer the following questions. Prove that 11"-6 is divisible by 5…

A: See the explanation below and follow the same if you understand the explanation please give ratings.

Q: Is the following statement true or false? For every integer n, if n is prime then (-1)^= -1. Which…

A: The given statement is: For every integer n, if n is prime then -1n=-1. We know that a non negative…

Q: Show that for every positive

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Q: Use mathematical induction to prove that 3"7" - 2 is divisible by 8 for every non-negative integer…

A: Initial case:

Q: Use mathematical induction to show that 32n – 1 is divisible by 8 for all positive integers n.

A: Solution

Q: Use induction to prove that 5^(2n) − 4^n is divisible by 7 for all natural numbers n.

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Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

Q: Show that if n is an integer and n³ + 5 is odd, then n even using a) a proof by contraposition. b) a…

A: Proofs: The proof is a valid argument that establishes the truth of a statement. Types of Proofs:…

Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: Use induction to prove 23n-3n is divisible by 5 (for all natural numbers n > or equal to 1)

A: To prove 23n – 3n is divisible by 5 (for all-natural numbers n ≥ 1) using induction method.

Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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Q: Observe that 1 = 1, 1 − 4 = −(1 + 2), 1 − 4 + 9 = 1 + 2 + 3, 1 − 4 + 9 − 16 = −(1 + 2 + 3 + 4), 1 −…

A: Note: Since you have asked multiple question, we will solve the first question for you. If youwant…

Q: Let p be a prime number such that p> 2. Use the proof by contradiction to show that at least one of…

A: See the solution attentively I think you will understand easily.

Q: Prove by Mathematical Induction that for any natural number n the number 52n+1 – 22n+1 is divisible…

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Q: 9. Ifn is a squarc-frce integer, prove that r(n) = 2" , where r is the number of prime divisors of…

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Q: True/False: Every integer greater than 1 is divisible by a prime number. True False

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Q: Use proof by contraposition to prove this.   If n is an integer and 3n is not a multiple of six,…

A: We prove that by contraposition If n is an integer and 3n is not a multiple of six, then n is odd.

Q: 1. Rehan proved by mathematical induction that for all the positive integers, n + 2n is divisible by…

A: Given that , For all positive integers, n3+2n is divisible by 3. To find an integer counter example:…

Q: 13. Pier Fermat (1601 – 1665) conjectured that any number F, of the form 2²" + 1 where n is…

A: Fermat number is Fn=22n+1

Q: (Determine if 1781 is divisible by 3, 6, 7, 8, and 9. )

A: Note: - Since you have asked multiple questions, we will solve the first question for you. If you…

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

Q: 2. The Hellenistic mathematician Nicomachus studied the following concept related to perfect…

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Q: Provide the smallest integer value that makes the following statement true: Statement. Every integer…

A: Fundamental Theorem of Arithmetic :- It states that every integer greater than 1 can be represented…

Q: P(n): 4" – 1 is divisible by 3.

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Q: Define an integer k to be odd if k − 1 is even. Write up a formal proof of the following, using an…

A: Consider the following information and prove the following:

Q: Prove the given statement: If the product of two integers is not divisible by an integer n, then…

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Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

Q: Prove that every positive integer greater than 1 has a prime divisor. b.) Prove that there are no

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Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: Use mathematical induction to prove that the statement, (a/b)n = an/bn is true for every positive…

A: Use mathematical induction to prove that the statement, (a/b)n = an/bn is true for every positive…

Q: Use proof by cases to show that if n is an integer then n^2 ≥ n.

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Q: Fill in the blanks in the following proof: Theorem 1. The set of prime numbers is infinite. Proof.…

A: Complete the proof of the given theorem as follows.

Q: A. Use the principle of mathematical induction to show that the tements are true for all natural…

A: Follow the procedure given below.

Q: Using the direct proof method, prove that for every integer a, if (a-2) is divisible by 3, then…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.