DISCRETE MATHEMATICS  Prove that for any integer a, the integer a(a + 1) is even. Use ANY METHOD of proof

Q: Using Direct proof, contrapositive, or contradiction: Proof that if x is an integer, then x2 + 3x…

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Q: Using indirect proof:  Let m be an integer. Prove that if m is odd then m² is odd.

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Q: Prove the bi-conditional: prove that if 5n+2 is even, then n is even, where n is an integer.

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Q: Using mathematical Induction prove that 3^(2n) -1 is divisible by 4 whenever n a is positive integer

A: Given term is 32n-1

Q: Prove that for every natural number a there is a multiple of a consisting of only 1's and O's.

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Q: . Find the sum of all integers between 100 and 500 that are exactly divisible by 12.

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Q: Advance Math; Discrete Structures; Proofs Prove that if mn and m + n are even integers, then both m…

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Q: Prove that if a and b are positive integers such that a|b and b|a, then a = b.

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Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

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Q: a(a?+2) 1) Prove that is an integer for a21. 3 2) Find the gcd(56,72)

A: 2. To find, gcd(56, 72) : 72 = 56×1 + 16 56 = 16×3 + 8 16 = 8×2 + 0 So, gcd(56, 72) = 8

Q: Discrete Mathematics* Prove that is a and b are nonzero integers, a divides b, and a + b is odd,…

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Q: proof by direct proof odd integer In is an then na is odd

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Q: Proof by contradiction that if n is a positive integer, then n is odd if and only if 5n + 6 is odd

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Q: Axiomatic ProofsProve a*(b-c)=a*b - a*c for all integers a, b, and c

A: The below expression is known as distributive property: -a*(b – c) = (a*b) – (a*c) for all integers…

Q: Proof by Induction 1 Prove by induction that 1 + 2-3 1 + n(n +1) for all integers n >1. 1.2 n+1

A: The given series is 11·2+12·3+...+1nn+1=nn+1 for all integers n≥1. Solve the series by induction.

Q: 1. Using proof by contradiction, prove Let n be an integer. If n- is even, then n is even.

A: Given: Let n be the integer.  To prove : If n2 is even, then n is even.

Q: USING DIRECT PROOF Let m be an integer. Prove that if m is odd then m² is odd

A: topic - numbers

Q: Let p be a prime number. Prove that (p -1)! a power p + (3p power2  +1)a is divisible by p, for any…

A: Consider the provided question, (a)

Q: proof by dlivect proof 米 IEn is an odd integer then nh is odd

A: Given- If n is odd integer. To Prove- That n2 is also odd integer. Concept Used- If n is an odd…

Q: 5) Using proof by contradiction prove the following implication. If n is an integer and 3n +2 is…

A: Given n is integer 3n+2 is even to prove n is even  We will prove by contradiction , suppose n is…

Q: gcd(a, 1) = 1 for any integer

A: True.

Q: 85. (a) Prove that any integer is expressible in the form 4n, 4n + 1, 4n + 2, or 4n + 3 where n is…

A: Here we use mainly dividend = divisor × quotient + remainder.

Q: Prove that there exists exactly one prime of the form n° -1. (When we say that a number p has "the…

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Q: Using a proof by contraposition prove that for any integer n, if n² is even then n is even.

A: The objective is to prove that n2 is even implies that n is even. So we prove the theorem by…

Q: Determine all integers n ≥ 3 such that Cn is planar.

A: ANSWER :

Q: Using Contrapositive Prove that for any integer a, the integer a(a+1) is even

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Q: Use Mathematical Induction to prove that the statement is true for all positive integers n. 3n+1 – 3…

A: Given that: 3+32+33+...+3n=3n+1-32

Q: Prove that √3 is irrational.

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Q: onsider the theorem: “for any integer x, if x mod 6 = 5, then (x2-2x+4) mod 4 = 3.” (a) Give an…

A: We recall some definition regarding modulo arithmetic and solve this problem. The detailed steps are…

Q: Mathematical Induction. Prove that for every positive integer n>3,  3^n > n^3.

A: Prove that for every positive integer n>3,  3n> n3.

Q: Proof or support on BINARY or NOT BINARY: ab=; on the set of integers. -4

A: On the set of integers, a*b=ab. It is NOT BINARY, for example 24=12∉Z

Q: *proof by divect proof If n is an odd integer then no is od

A: Direct proof

Q: Question: Prove that by mathematically Induction 2" < (n+ 1)!, for all integers n2 2.

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Q: Axiomatic ProofsProve -(-a)=a for all integers a

A: We need to prove that, for all integers a,

Q: DISCRETE MATH use ANY METHOD of proof. 1. Prove that if a and b are positive integers such that a|b…

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Q: Question Prove that for all positive integers a and d, if d|a then d is less than or equal to a.

A: Solve the problem

Q: 3. Prove: For all positive integers, n, n(n + 1)(n + 2) >i(i + 1) = 3

A: Kindly find the attachment

Q: square of any odd integer is od.

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Q: Use mathematical induction to prove that the statement for all positive integers. + (2n-1) = n² a)…

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Q: contrapositive proof. Include explanation.

A: Given that if n is an even, then 7n + 4 is an even integer.

Q: 2""(-1)" +1 2. Prove that 1-2+2² – 2³ +...+(-1)" 2" =- for all positive integers n.

A: We prove this result by using mathematical induction. For answer please look at step 2.

Q: (f) Give a recursive definition of a", where a is a nonzero real number and n is a nonnegative…

A: Solution

Q: II. Prove the following statements using the Principle of Mathematical Induction (PMI). an+! 1.) Let…

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Q: Prove that If a and y are integers and r + y is odd , then a – y is odd. I Show that for any integer…

A: The only way to get an odd sum or difference of two integers is if one integer is odd and the other…

Q: 12. Prove the Fundamental Theorem of Arithmetic: Every integer greater than 1 is a prime or a…

A: The objective is to prove the Fundamental Theorem of Arithmetic, that states: “Every number greater…

Q: Discrete Mathematics: Prove that every prime number greater than 3 is either one or more or one less…

A: We need to prove that any prime number which is greater than 3 is either 1 more than or 1 less than…

Q: Using direct, indirect and contradiction 1. Prove that if n is a positive integer, then n is odd if…

A: To Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.

Q: Proof or support on BINARY or NOT BINARY: a *b = a – b on the set of positive integers.

A: Given the operation a*b = a-b  It is not binary operation on the set of positive integers For…

Q: II. Prove the following statements using the Principle of Mathematical Induction (PMI). an+1 - a 1.)…

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Q: Discrete mathematics for comuter science. Answer + why Let Xi = {2i - 1}. Then Uni=1 Xi is a) a…

A: Given Xi={2i-1} now by putting some values of i = 1, 2, 3, ......., we have X1={1}, X2={3}, X3={5},…