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

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: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

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

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Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: Prove that there are no positive integers a and b such that a + b³ = 250.

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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: Let a and b be two integers. Consider the following statement: If a+b= ab, then a and b must both be…

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

A: Given statement is false because

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

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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: b) Prove by contradiction: There exist three consecutive odd integer a, b, and c such that a? + b² =…

A: Explanation: Given that,  3 consecutive odd integers a, b & c. Let us assume m & n are…

Q: Let a, b, and c be natural numbers. Prove a ∙ (b + c) = a ∙b + a ∙ c.

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Q: Determine whether the statement is true or false. Prove the statement directly from the definitions…

A: a. For all integers a,b, and c, if a | bc, then a | b or a | c.To demonstrate this assertion, we'll…

Q: Show that if a and b are coprime integers, then gcd(a + b, a² – ab + b²) = 1 or 3.

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Q: let a b c and d be integers, prove if a|b and c|d then ac|bd

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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: let a, b, and c be integers. if a divides c and b divides c, must a*b divides c?  Prove or come up…

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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 this statement For all integers a, b, and c, if a ∤ bc, then a ∤ b

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

A: Let x and y be two odd numbers given by; 2m+1,2n+1 where m and n are integers;

Q: Prove by contradiction that for all integers a, b, and c, if a|b and a | c, then a |(b + c).

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Q: Prove that if a and b are both odd integers, then 8 (2a2+36²).

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