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

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

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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 that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: Use a direct proof to show that if a and b are positive integers, then ab+ba22.

A: Answer

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

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

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: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Let a, b be two positive integers with (a, b) = 1. If ab = c, then prove that there exists

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Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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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.

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

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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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: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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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: contradiction

A: Please refer to the image below

Q: Suppose that a, b, and c are integers. Prove that if a2 +b2 =c2 then a or b is even.

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Q: Show that is a and b are positive integers such that (a | b) and (b | a), then a = b

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Q: Prove or disprove. There are integers a, b, and c such that a divides bc, but a does not divide b…

A: We have to prove or disprove that if integers a, b, and c such that a divides bc,but a does not…

Q: let a, b, and m be integer, prove that if 2a+3b>=12m+1, than a>=3m+1, or b>=2m+1

A: To prove: For integer a, b, and m if   2a+3b≥12m+1Than, either a≥3m+1, or b≥2m+1.

Q: If α is even, prove that α-1 is even. If α is odd, prove that α-1 is odd.

A: We know that the multiplication of 2-cycles can help us find any permutation of a finite set, so we…

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: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: Show that if a and b are positive integers, then ab = lcm(a, b) . gcd(a, b).

A: Consider two cases: 1. a and b are multiples of n i.e. they have a common factor n So a=nx, b=ny…

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

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Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: Prove that if a is an integer and b is a positive integer, then exists unique integers q and r with…

A: Please check further steps for solution.

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

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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 Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

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: Suppose a, b, and c are integers and suppose a |b and ałc, where a{c denotes that c is not divisible…

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Q: Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

A: We have to prove the following statement -  If a,b, and c≠0 are integers, then a∣b if and only if…

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: Prove that √2 is not an element of Q(?).

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Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

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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: 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 a and b are integers, with b > 0, then there exist unique integers q and r satisfying…

A: To Prove -  If 'a' and 'b' are integers, with b>0, then there exists unique integers 'q' and 'r'…

Q: a2 − b2is divisible by 5.

A: First observe that one's place of the square of a positive number can only one among 0,1,4,9,6, and…

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

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

A: Number system.. properties

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.

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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