Prove that if a is odd and b is odd, then (a + 1) b is even

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: Prove thq+ for all integer a, b, and c if 9+b of a and k is even.

A: Introduction: Even and odd numbers are also referred to as: Even Number: An even number is one that…

Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: Prove the theorems: i) "If n is an integer, and if n² is even then n is even". ii) "If n is an…

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Q: Use a proof by contradiction or by cases to show: No integers y and z exist for which 2y2 + z2 = 14

A: Result: Product of two positive term is always positive and product of two negative term is always…

Q: Prove the following by contrapositive: If x is an even integer, then x² is even.

A: Let x is not even, it has to prove that x is not an integer or x2 is not even. Let x is not even…

Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: 6. Prove that if x, y, and z are integers and x + y + z is odd, then at least one of x, y, and z is…

A: Given: x , y , and z are integers and x + y + z is odd.  We have to show that at least one of x , y…

Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

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Q: Let a and b be positive even integers. Prove that (a, by 2

A: Here, a,b denotes the gcd of a and b. a2,b2 denotes the gcd of a2 and b2. Given that a and b are…

Q: If r and s are real numbers and r < s, then there exists a positive integer t such that r+t = s.…

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Q: Let a, b eN. Prove that if a + b is even, then there exists nonnegative integers x and y such hat x²…

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Q: •Prove: For all integers a, b, and c, if a|b and alc, then a|(b+ c). we can assume a #0.)

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Q: Let x and y be integers. Prove that if x2 + y2 is odd, then x+y is odd

A: Given x and y be an integers. We have to prove that if x2+y2 is odd then x+y is odd. Since given…

Q: Prove the following statements: (a) If two integers are both odd, then their sum is even. (b) If two…

A: Consider two odd integer

Q: Prove that for every positive integer m, there is a positive integer k such that k does not divide m…

A: Given:-                 prove that every positive integer m, there is a positive integer K such that…

Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Let x and y be integers. Prove that if x2 - y2 is even, then x + y is odd

A: The given statement looks wrong. If two numbers x and y are integers and x2-y2 is even  Now the…

Q: Prove: If a, b and c≠0 are integers, then a|b if and only if ac|bc.

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Q: Let x,y,z∈ Z be integers. Prove that if x(y+z) is odd, then x is odd and at least one of y or z is…

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Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

Q: For every pair of integers x and y, if x – y is odd, then x is odd or y is odd.

A: Let x-is even integer and y-is even integer. x=2m, y=2n

Q: Find the generalized union and intersection: 3) A = {1,2,3,...} A₂ = {x|x = 22m where m eset of…

A: Given that, Λ={1,2,3,…} Λλ=x|x=2λm where m∈set of integers from -λ to +λ The objective is to find…

Q: Prove that if m is a positive integer and r is a real number, then Lma) = l=] + -+ + + [-+ + r+ m

A: We are given a Inequality involving the sum of floor functions, and we have to prove that.

Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: x]+n x+n m m

A: By using definition of greatest integer function, For x∈ℝ we have, x=x+x, 0≤x&lt;1 where, x is…

Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: Let x and y be integers. Prove by the contrapositive that if x2 - y2 is even, the x + y is even

A: Let x and y be integers such that x2-y2 is even. We have to prove that x+y is even. Suppose that x+y…

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

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Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

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

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

Q: 1. If a is an odd integer, then a2 +3a+5 is odd. (use direct proof) 2. Suppose x and y are even. If…

A: Given:1. If a is an odd integer , then a2+3a+5 is odd2. Suppose x and y are even.If x i even, then…

Q: Prove that if the product of two integers qs is odd, then q2 + s2 is even. Then show that for…

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Q: Prove that if ∣A∣ = 1 and all entries of A are integers, then all entries of ∣A−1∣ must also be…

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Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

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: Prove that x + 3 is odd if and only if x² is even.

A: Proof: x+3 is odd  Let x+3 = 2k+1

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: Let x be positive integer. Prove If x2 + 1 is even, then x3 is odd

A: Consider the provided question, Let x be positive integer. We need to Prove that, If x2+1 is even,…

Q: prove if a and b are both odd integer, then 16/ (a^4) + (b^4) -2.

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Q: Prove that for integers a and b, (a -3)b2 is even if and only if a is odd or b is even.

A: We need to prove that for integers a and b, (a -3)b^2 is even if and only if a is odd or b is even.

Q: 6. Prove that for all integers a, b, and c, if a² + b² = c², then at least one of a and b is even.

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Q: 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 ifx^2−6x+ 3is odd, then x is even

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

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Q: and

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Q: Prove that if a, b and c are positive integers such that a divides b and b divides c, then a divides…

A: It is given that a,b and c are three positive integers. It is also given that a divides b. Then it…

Q: Let a and b be real numbers. Prove that if 4a+16b= 123, then a and b are not integers

A: Topic :- Equations In the given solution below 123 is not completely divisible by 4.