Prove: there exist no integer a and b for which 21a + 30b = 1.

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

Q: Prove that the statement is true for every positive integer n.

A: Given, 3+9+15+..+(6n-3)=3n2

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: 6. Let r> 2 be an integer. Show that R(2r, 2r) > 2

A:

Q: 5. Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive integers.

A:

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

A:

Q: Prove that 5 does not divide 5n+3 for any integer n.

A:

Q: Prove, using the definition of odd, that for all integers r, that 6r+5 is odd.  b) true or false:…

A:

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: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A:

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

A:

Q: Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 =…

A: first we take a odd integer then we write in 2y+1 form then we transfer to difference of two…

Q: (2) Prove that there are no integersm and n such that 7m + 2ln - 15.

A: There are no integers m and n such that 7m+21n=15

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

A: Proving a statement by contradiction means assume the negation of statement true and prove the…

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

A: answer

Q: Prove by mathematical induction that 3n+2n+3 is divisible by 4 for all positive integers

A:

Q: Prove that the statement is true for every positive integer n.

A: Prove that the statement is true for every positive integer n.…

Q: b) Prove by contradiction: There exist three consecutive odd integer a, b, and c such that a? + b² =…

A: Explanation: Given that,  3 consecutive odd integers a, b & c. Let us assume m & n are…

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: 3. Find integers s and t so that 1 7s + 11t. Show that they are not unique.

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: Let k be some positive integer. Prove that a number of the form 4k +3 can never be the sum of two…

A:

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: Use the contradiction method to prove: There exist no integers a and b for which 21a + 30b = 1

A:

Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: b) Hence or otherwise prove that there do not exist three consecutive integer values of n for which…

A: First let us assume that 29n+5 is a perfect square for some consecutive integers n and n+1. Then…

Q: Prove that there are no integers m and n such that m2=4n+2.

A:

Q: Prove that the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: Prove that every integer greater than 27 can be written as 5a + 8b, where a, b e Z+.

A: Let P(n) be the statement, "For n∈ℕ there exist a, b>0 such that              n+27=a·5+b·8 Use…

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: Prove that 4 does not divide n2 +2 for any integer n.

A:

Q: Consider the statement: "there are no integers and such that 20a+4b%3D1." Is this true or false?…

A:

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

A:

Q: Prove that 3/3 is irrational.

A: A number which can be written in the ratio , where and are integers are called rational numbers.…

Q: Use proof by contradiction to show that for all integers m, 7m+4 is not divisible by 7.

A: Recall the following fact about divisibility.

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

A:

Q: Prove that n2 + 4 is not divisible by 3 for all integers n.

A:

Q: Prove that 3+√3 is irrational.

A: As per guidelines we are supposed to solve the maximum one questions at  a time kindly post another…

Q: Prove {12a + 4b : a and b are integers} = {4c : c is an integer}

A:

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

A:

Q: b. Prove that 2 is irrational.

A:

Q: Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

A:

Q: Suppose that a, b. are two integers such that 3a² – b = 1. Does it follow that a and b are…

A: Given that,

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a non-negative integer

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

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

A:

Q: Prove that 3" >n² for all positive integers

A:

Q: such

A: Given: 165x2-21y2=19

Q: Prove that A,(3, 2) = q² for any integer q>2. %3D

A: To prove : Aq3, 2=q2 for any integer q≥2 Pre-requisite : 1. Sphere-covering bound : For an integer…

Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

A: