Prove that 2 divides a if and only if 2 divides ao. Prove that 4 divides a if and only if 4 divides ao + 2a1. Prove that 8 divides a if and only if 8 divides ao + 2a1 + 4a2.

Q: Find the antiderivtive: /102³-52² +7 x6 -dx = + C

A: since you have posted multiple questions in this question which are not interlinked so i have solved…

Q: If (√3+i) 100 = 29⁹(a + ib), then show that a² + b² = 4

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Q: A positive integer is divisible by 5, but it leaves remainders of 1 and 7 (resp.) when divided by 3…

A: Here we will use Chinese Remainder Theorem to find solution.

Q: 1) Show that the square of an odd integer is odd. 2) Show that the product of two odd integers is…

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Q: Show that the first statement is true while the second statement is false. Statement 1: There exist…

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Q: Find /( - 32² + 02 - 3³ + √²) dz 6 3x6 x6

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Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: 19) Determine a polynomial that represents the area of a square having sides of length s x + 8. B)…

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Q: 27. (LO14) Divide x³+ x² 4x + 9 by x + 3 to find the quotient and remainder A)x² +4x+ 8; R= 33 C) x²…

A: To divide the given polynomial

Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: 5. if we have two polynomials pl=3x5 + 8x* – 2x2 + 4x + 1 p2= 2x6 + 3x* – x² + 4x Find:- Your answer…

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Q: If P, be a polynomial of degree n, i.e., P, = aok" + a;k"-1. + a, then A" P. = aon!.

A: We have to solve given problem:

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: 20) Determine a polynomial that represents the area of the figure. 20) 3x - 5 3x + 5 A) 9x2 - 25 B)…

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: prove that the sum of three odd integers is odd using indirect method. show your solution and…

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Q: Prove that √1+i√3+√1-i√3 = √6

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Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: Show that if x is an integer, then gcd(x + 1, x2 – x + 1) divides 3. -

A: First, we find gcdx+1, x2-x+1 Now, factors of x+1 are x+1, 1 x2-x+1 is a quadratic equation of the…

Q: Expand the expression for Mr Taylor.

A: The previous solution is incorrect. Please follow this one.

Q: 7. f6x4-2x³-21x² + 7x + 8 is divided by 3x - 1 to give a quotient of 2x³ – 7x and a remainder of 8,…

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Q: Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4.

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Q: • Show that if m divides 3, then m divides 9. (We say a divides b if there exists an integer k such…

A: Let  m divides 3 .  Then there exists an integer k such that 3 = mk. Now, 9 can be written as…

Q: Find all sollutions in integers of 34x+8y=6

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Q: y-1 = x²y-xy-x²+x f(x,y) (i) f is continuous at every point of its domain (ii) the domain of f is…

A: Let's analyze each statement one by one:(i) f is continuous at every point of its domain.To…

Q: if two even integers are multiflied, then the product is also even

A: Ans. True

Q: 1) Solve the O.DE. qiven below y'+ 3yt= tye"

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Q: If GCD of (a,b) is 1 we can find the lcm by multiplying a and b. True or false

A: if GCD(a,b) is 1, We can find lcm by multiplying a and b.   This statement is True.

Q: Show that the system consisting of all elements of the form a+b√5, where a and b are any rational…

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Q: Show that if 9 | a3 + b3 + c3 then 3 divides at least one of the integers a, b, or c.

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Q: Find the minimal polynomial for 3√ 2 + 3√ 4 over Q.

A: Let x=23+43. To Find: Minimal polynomial for 23+43 over Q.

Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.