Prove that if a > 2 and b are integers, then a † b or a { (b+1).

Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

A:

Q: Prove that 1– 2+2² – 2° + -.-+(-1)"2"=2**1 for all nonnegative integers n.

A:

Q: Use mathematical Inductlon to prove that for r>0 and any positive Integer

A:

Q: Prove that the set of positive integers is equal to the set {1,2,3,...}. (Using Axioms)

A: This is a problem of set theory and cardinality.

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

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: What is the set Z+ of all positive integers?

A: Set of integers are represented as: ℤ={.....,-3,-2,-1,0,1,2,3.........} Set of natural numbers:…

Q: Find all integers z, positive or negatives, such that z2 + z is prime.

A:

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Prove that the K4 is not consecutive.

A:

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: 6"- (-4)" %3D k 1 k odd IM- 2]

A:

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

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: 1.(a) Do there exist integers a, b, c such that gcd(a, b) = 30, gcd(b, c) = 24, gcd(a, c) = 54?…

A: (a) gcd(a, b) = 30       gcd(b, c) = 24       gcd(a, c) = 54      =&gt; 30 | a, 54 | a           24…

Q: () is even. 3. Prove that for all positive integers n, the number

A:

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: Abstract Algebra

A: To find non-negative representations under the given conditions

Q: -ん 2 kodd

A:

Q: Prove that for any positive integers m and n there are m consecutive positive intege numbers such…

A: As per the question we have to prove that for any positive integers m, n there exist m consecutive…

Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Discrete

A:

Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

A:

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

A:

Q: prove that in any eight composite positive integers not exceeding 360, at least two are not…

A: Prove that in any eight composite positive numbers not exceeding 360, at least two are not…

Q: prove that for all integers m>2,m^2+m-6 is composite

A:

Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

A:

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: Prove that the subset R/Q , the set of all irrational numbers , of R id of second category .

A:

Q: Determine (with proof) if there exist integers m and x such that 1997m + 47x2 = 3(17x – 1). | If…

A: Given equation is 1997m+47x2=317x-1. To determine: Whether integers m and x exists or not.

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: (2) Prove the statement "Every positive even integer less than 16 can be written as a sum of three…

A: Perfect square: n is a perfect square, if and only if, there exist an integer k such that n=k2.…

Q: prove that √2+√3 is irrational

A: For rational we needed form p by q P and q are integer

Q: Suppose there is a currency which only uses 3-cent coins and 7-cent coins. Prove that for all…

A:

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  =&gt; k | {(n+4) - (n-1)} =&gt; k | 5

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

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

A:

Q: Prove by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n V3n

A: For all positive integers n, we need to prove that, 12·34···2n-12n&lt;13n

Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: 16. Prove that, for integers a and b, gcd(a, a + b) gcd(a, b). [Hint: Mimic the proof of Lemma…

A:

Q: Let a and b be integers, not both zero. Then a and b are relativelyprimeif and only if 1 = ax + by,…

A: Two integers are relatively prime when there are no common factors other than 1. This means that no…

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

A:

Q: Let a relatively prime positive integers. If there exist integers s and t with as + ht t - 11 then

A: if gcd (a,b) =d then there exists integers m and n such that d = a m + b  n  and d is the smallest…

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Prove that the set of irrational numbers in [0,1] is not countable.

A: The objective is to prove that the set of irrational numbers in 0,1 is not countable. Let's prove…

Q: Prove that (1 -)(1-÷)(1-→).--(-) for all integers m 2 2 by m 2m nathematical induction.

A: *concept:- formula, a^2 - b^2 = (a+b)*(a-b).

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

A: