Given n is a positive integer, prove that n is even if and only if 3n + 5 is even. You must use definitions of even (and odd, if needed). Do not use “even plus even is even” or similar rules of thumb.

Given n is a positive integer, prove that n is even if and only if 3n + 5 is even. You must use definitions of even (and odd, if needed). Do not use “even plus even is even” or similar rules of thumb.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 10E: Let be a nonzero integer and a positive integer. Prove or disprove that .

See similar textbooks

Related questions

Q: Which of the following statements is (are) true for all positive integers? 1) n' + 2n is divisible…

Q: By dragging statements from the left column to the right column below, construct a valide proof of…

A: To prove the below statement: For all integer n, if 9n is even, then n is even

Q: Show that for all positive integers n, 13 + 2° + ...+ n³ = (1+2+3+ ...+ n)?

Q: Find a positive integer n for which the equality holds = 1 2

Q: Given ? is a positive integer, prove that ? is odd if and only if 7? − 5 is even. You must use…

Q: Provide a proof by contradiction of the statement below. For all integers x, 3x + 2 is not divisible…

Q: Use direct proof to prove the following statements. 1. If x is an odd integer, the x2 +3x +5 is odd.…

Q: Use the definition of even to verify that the expression is an even integer. 12m + 4n+10 2-10m

A: An even integer is an integer which is divisible by 2. Let a is an even integer i.e. a is divisible…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: Prove by contrapositive. Let k be an integer. If 5k+9 is odd then k is even.

Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a > 0, and b < 0. Prove that xn + ax + b = 0 has a unique root

Q: show that for eury A s integer nz3; A n- Am-2 +A2J. Mence determine ASO.

A: To show: That An=An-2+A2-I for n≥3 then find A50. Given: The matrix is A=100101010. Calculation:…

Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

Q: Let n∈Z. Prove that x is odd if and only if 5x + 4 is odd. Identify the type of the above…

Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is P(0)…

A: For each positive integer n, let P(n) be the property 5n − 1 is divisible by 4. Write P(0). Is…

Q: Consider the following statement. Is there an integer that has a remainder of 2 when it is divided…

A: The statement " Is there an integer that has a remainder of 2 when it is divided by 5 and a…

Q: Write a short proof of the following and give a counterexample If n is an integer, then 2n + 1 is…

Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

Q: Use contradiction to prove: if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer and let n is even therefore, n=2k for some integer k therefore,…

Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

Q: Prove or Disprove: If x is an integer and 7x³+9 is even, then x is odd.

A: According to the given information, it is required to prove or disprove the given statement.

Q: Use a proof by contrapositive to prove that, if (x + 12)/x is an integer, then x < 13.

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: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c & b/c , we know there exists integers K1 & K2 ,with K1 a=K2 b=c--1Also…

Q: Which of the following statements is (are) true for all positive integers? 1) n' + 2n is divisible…

A: given statement 1) n3+2n is divisible by 32) 52n-1 is divisible by 24

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Let a and b be integers. Proof by contrapositive that a + b is an even number if and only if both…

Q: For integer values of a, use a proof by contraposition (also called contrapositive or indirect…

Q: Prove the statement for positive integer n and r, with r ≤ n. P(n, 1) = n

A: To prove the statement

Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: Let x be an integer. Prove or provide a counterexample: If 4x+7 is odd, then x is odd.

A: We provide a direct proof and give counter example to show that given statement is false.

Q: Prove, using any method, the following statement: if n is an integer, then (3n+1)(5n+4) is an even…

Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: . For a positive integer a, determine whether the following statements are alw times true, or never…

A: we have to check

Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: prove n2 + 5 greater than or equal to 4n for all integers n

Q: Prove that if n is an integer, these four statements are equivalent: (i) n is even, (ii) n + 1 is…

Q: For all integers a, b and c, if a divides b and a divides c, then a divides (4b - 3c). True False

A: Answer is TRUE

Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation

Question

Given n is a positive integer, prove that n is even if and only if 3n + 5 is even.
You must use definitions of even (and odd, if needed). Do not use “even plus even is even” or
similar rules of thumb.

Expert Solution