Prove that 3n^2 + 1 is even if and only if n is an odd integer.

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: prove or disprove: the product of any odd number and any even number is even.

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Q: Suppose that you know that a golfer plays the first hole of a golf course with an infinite number of…

A: To prove that the golfer plays every hole on the course when given that a golfer plays the first…

Q: Use a direct proof to show that the product of two odd integers is odd.

A: Introduction: A direct proof is a series of statements, the final of which is the conclusion to be…

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Given that it is to be found all three-digit numbers with different digits with the property that…

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: Use mathematical induction to prove that +1+2+4+...+2"-² = 2"- for all 2 -- positive integers . п.

A: We need to prove using mathematical induction given statement First we will show statement holds for…

Q: 2"-1 for positive integers n.

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Q: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: Determine whether the statement is true or false. Justify your answer with a proof by Direct Method…

A: The product of any two integers is odd. - True

Q: Prove that the summation of three odd integers is odd.

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Q: Prove that 1+6x-x is always a positive real number if x =3+Ai for some real 33 number 4

A: The given number is  1 + 6x - x2

Q: How do I find f prime of 3 if f(x)=13x/4+x2

A: Given,           fx=13x4+x2

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: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

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: Let n and k be positive integers with 2k + 1 ≤ n. Find the number of ways to choose k children from…

A: Consider the given: Let n and k be positive integers with . Choose the number of ways in which k…

Q: 9. Prove that there are as many positive real numbers as there are real numbers.

A: Consider the set X and Y. Then we said that |X| = |Y| if there exists bijective function from X to…

Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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Q: 3) Prove that there do not exist two rational numbers whose sum is irrational.

A: To prove: There do not exist any two rational numbers such that their sum is irrational.

Q: 4. Suppose we are in a parallel universe where 6 dollar coins and 11 dollar coins exist. Show that…

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Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

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

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Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

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Q: All Fermat's number is clearly odd numbers

A: Fermat number.

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: 27. Show that if x and y are positive real numbers that have the same first n digits in their…

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Q: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

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Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

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Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

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: sum of two even integers is even

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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: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: (2) Explain why the rational numbers are those having periodic representation or with- out infinite…

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Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

Q: Question 1. Show that if seven integers are selected from the first 10 positive integers, there must…

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

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Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Consider the provided question, we need to prove that, Any positive integer N is divisible by 11 if…

Q: 3. Prove that 0 divides an integer a if and only if a 0.(In your proof. you can freely ILNE your…

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Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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