Every positive integer is always divisible by 1 and by itself. That is, for every posi- tive integer n we always have 1|n and n|n. We say an integer p ≥ 2 is a prime number if its only divisors are 1 and p. (By convention, 1 is not a prime number.) (a) List the first ten prime numbers. (b) If a positive integer n is not prime, then it is called a composite number. Show that every positive integer n ≥ 2 can be written as a product of prime numbers. (This is called a prime factorization of n. Notice that some primes may need to be repeated. For example, 180 = 2² × 3² × 5 and the primes 2 and 3 are repeated.) Hint: Consider two separate cases based on whether a given number n is prime or composite. For composite numbers, suppose by strong induction that all smaller integers have a prime factorization.

Q: The following “Theorem” is obviously not true. Explain what’s wrong with the proof. [5]  Theorem: 1…

A: We see in the proof that it is a fact that n2 > n when n > 1. That means n2 is larger than n…

Q: Let m, n be two integers. For each of the statements below, decide if it is TRUE or FALSE: (a) If…

A:

Q: The following “Theorem” is obviously not true. Explain what’s wrong with the proof. Theorem: 1 is…

A: Given Data: Given Theorem is : 1 is the largest natural number. This theorem is not true. We have to…

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: • for any integers a, b, c, • n is not prime and p6 |n, where p; is the ith prime (i.e., p₁ = 2,p2 =…

A:

Q: Show that for every positive

A:

Q: How do I prove: If n > 1 is a natural number then there is a prime number p such that p|n.

A: This is a theorem of real analysis in which we have to find that there is a p who divides n.

Q: How do I prove: Let p be a prime number. Show that for any integer a the greatest common divisor of…

A:

Q: Prove the statement below directly from the definition of divisibility. For all integers a, b, and…

A: Suppose a|b and a|c. This means that there are numbers f and g such that af=b and ag=c.

Q: For every integer n, n is even if and only if 3n - 11 is odd. (a) Notice the Result is a…

A: The given biconditional statement is, " For every integer n, n is even if and only is 3n-11 is odd".

Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: Prove or disprove: If an integer is divisible by 6 and by 10, then it is divisible by 60.

A: Answer

Q: Is the following proposition true or false? Justify your conclusion with a proof or a…

A: Check the following proposition, "For each natural number n, if 3 does not divide (n2+2) then n is…

Q: 5. The set 5 contains some real numbers, according to the following three rules. (i) is in S. (ii)…

A: First, let's prove that all elements in S lie in the interval [1/2, 1].1. By rule (i), 1/1 is in S.…

Q: 1. Is the following statement true? If yes prove. If no give a counter example. If a and b are any…

A:

Q: If A = {x : x is a natural number },               B = {x : x is an even natural number}  C = {x : x…

A:

Q: Prove the following:For any natural number n, if n 2 + 6 n is an even number, then n is an even…

A: Consider the following information and prove the following:

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: Given that n is an integer, it follows that 2n is an even integer. O a. Inductive, strong. b.…

A: ~ A deductive argument is an argument for which the premises are offered to provide logically…