1 When two even integers are multipliec, the resulting product is also an ever integer. Answer:Blank 1 Caunterexample:Blank 2 Blank 1 Adu your answer Blank 2 Ada your answer

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

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Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that N × N × N is countable. Does your argument generalize to the Cartesian product of k…

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Q: b.) Show that if a and b are positive integers where a is even and b is odd, then (a,b) = (a/2, b).…

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Q: Prove that for a positive integer n we have that  33n+3 - 26n - 27 is divisible by 169.

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: 3. Let z and y be 3-digit numbers such that y is equal to r written backwards. For example, if x =…

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Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

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: If gcd(a, b) = 1 and c\a, then gcd(c,b) = 1. If d # 1 is the smallest positive divisor of an integer…

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Q: An integer n > 2 is composite if and only if n does not divide (n - 1)! + 1. :Select one True False

A: The solution is given as

Q: a and c are odd integers.  ab +bc is even for every integer b use indirect and direct proofs

A: Direct proof:- Given that, a and c are odd integers. a=2m+1, c=2n+1 Now,…

Q: 6. Prove by CVI that every natural number n > 2 is a product of prime numbers. NOTE. A prime number…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: If gcd(a, b) = 1 and c|a, then gcd(c, b) = 1. If d + 1 is the smallest positive divisor of an…

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Q: 5. Prove that the sum of the first n natural numbers is equal to n(n+ 1)/2, and deduce that the sum…

A: Sum of n number is given by

Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

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: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

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

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

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Q: 2. Prove that n = aŋɑp-1 aza,a, is divisible by 8 if and only if aza,a, is divisible by 8. A…

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Q: 1- The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit…

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Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: Show that there are infinitely many integers n such that 4n^2+1 is simultaneously divisible by 13…

A: We need to prove there are infinitely many integers n such that 4n^2+1 issimultaneously divisible by…

Q: 4. Let a, b, and d be integers such that dļab. Is it true that either da or d b? If you claim that…

A: Given integers a,b and d. We need to check whether if d/

Q: If a and b are integers with gcd(a, b) = d Then gcd (-- d.

A: Given a statement

Q: Prove or disprove: If three consecutive integers are multiplied together, and the second in order of…

A: We first look out some examples to understand the fact- choose three consecutive numbers be 1,2,3.…

Q: (1) Disprove the statement "For all integers a, b if a | b2 and a < b then a | b." by giving a…

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Q: 2. Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Prove that 3√2 is irrational. you may use the proposition that the product of two even integers is…

A: Note: we know that square of an even and an odd integer must be an even and odd integer…

Q: Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even

A: Given,  for any integer n the product n(n + 1) is even  We have to prove it using the method of…

Q: C) Prove that the product of k consecutive positive integers is divisible by k!

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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: A perfect cube is an integer x such that there exists a an integer y and x = y³. Prove that the sum…

A: Let n,n+1,n+2 are the three consecutive integers. consider n=1 therefore,  13+23+33=1+8+27=36 which…

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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

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Q: 10. Prove: A and B countable > Ax B countable.

A: According to the given information, Let A and B be two countable sets then it is required to prove…

Q: Find the number of four non-negative integer ordered tuples (x1, x2, x3, x4) such that xi + x2 + ¤3…

A: The Number of non negative integral solutions of x1+x2+…………+xn=r, where each of xi≥0 is given by…

Q: 9. One merchant has seven Asava horses (A), a second has nine Haya horses (H), and a third has ten…

A: Given, One Merchant has seven Asava horses (A), a second has nine Haya horses (H) and third has ten…

Q: 1. Prove: 11" – 6 is divisible by 5 for all positive integers, n.

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Q: 3. Let z and y be 3-digit numbers such that y is equal to x written backwards. For example, if x =…

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Q: Give a contrapositive proof of the theorem  If n is an integer and 3n+2 is even ,then n is even

A: We have to prove given statement by contrapositive proof.

Q: Two numbers a and b are chosen from the set 1,2,.., 3n In how many ways can these integers be…

A: In the question we have to find total no. of cases by which a2-b2 is divisible by 3.

Q: Prove that if the product of two integers ab is odd, then a2 + b2 is even

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Q: 1. Show that if then p and q have no non-trivial common factor. and q are natural numbers and there…

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Q: 2 There are integers m and n such that m² +n² is a perfect square.

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Q: Prove that if a is a positive integer, then 4 does not divide a2 + 5.

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Q: prove that if a>4 then (a-2)^2>4

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