(a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers. Prove that a (bx+cy).

Q: Prove that if alb and alc then a (bx + cy) for all integers x, y.

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Q: Find necessary and sufficient conditions on the numbers a, b, c E R under which the polynomial xª +…

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Q: Let R[r] be the set of all expressions 00 a = do+ajx+azx += i=0 where a¡ E R for all nonnegative…

A: We will be solving part (b) as mentioned. (b). Given ℝx be set of all expressions…

Q: Prove: If x, y, z are integers with x2 + y? = z², then either x or y (or both) is even. Hint: Do a…

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Q: Let x be an integer. Prove that 2|x and 3|x if and only if 6|x. Generalize and prove.

A: According to the given information, let x be an integer. It is required to prove that:

Q: x and y. Prove that if alb and alc, then a|(xb+ yc) for any integers

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Q: Find all zeros of the polynomial function. Ax) = x* - 2x + 8x2- 32x – 128 O -4, -4, -2, and 4 O -4,…

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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: Prove by any method(contradiction/contrapositive or cases) Prove that if a and b are positive…

A: Prove that if a and b are positive integers such that a|b and b|a, then a=b Let a and b are positive…

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: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: The fields Q(√3) and Q(11) are isomorphic. O True O False

A: We have to determine that whether the following statement is true or false ; The fields Q33 and Q113…

Q: Use a direct proof technique to prove the following theorems: For all integers r and y, r + y - 3 is…

A: We have to prove that , For all integers x and y , x2+y2-3 is not divisible by 4. We will have 3…

Q: Let be the following set of rational numbers: Consider the algebraic structure (R2, +, ·), where +…

A: Field :-  A commutative ring F with unity is called field, if each nonzero elements of F has…

Q: Prove that there exists two integers a and b such that a+b > ab

A: We want to prove that there exists two integers a and b such that a+b>ab

Q: Let V be all elements of R2 that have the form (a, a2) where a is a real number. Show that (3,9) is…

A: Let V be all elements of R2 that have the form (a, a2) where a is a real number. We have show that…

Q: Prove that there are integers x and y such that 10x-13y=1

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Q: Prove that if a and b are integers of the form 4n+1, then ab is of the form 4n+1.

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Q: (b) Let A and Bị be independent; similarly, let A and B2 be independent. Show that in this case, A…

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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: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

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Q: Prove that the set of numbers Q(√2) = { a+b√2: a, b = Q} is a field.

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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: Let a, b, and c be integers with a 0. Prove that if a | b and a | (b + c), then a | c.

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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: Use a direct proof technique to prove the following theorems: For all integers z and y, r² + y - 3…

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