A polynomial in Z[x] is said to be primitive if the greatest common divisor of its...

A polynomial in Z[x] is said to be primitive if the greatest common divisor of its coefficients is 1. Prove the product of two primitive polynomials is primitive. [Hint: Use proof by contradiction.]

Expert Solution

Colby Messinger answered 2 years ago

Java program: Greatest Common Divisor Write a program which finds the greatest common divisor of two...

Java program: Greatest Common Divisor Write a program which finds the greatest common divisor of two natural numbers a and b Input: a and b are given in a line separated by a single space. Output: Output the greatest common divisor of a and b. Constraints: 1 ≤ a, b ≤ 109 Hint: You can use the following observation: For integers x and y, if x ≥ y, then gcd(x, y) = gcd(y, x%y) Sample Input 1 54 20 Sample Output...

a complex number z is said to be algebraic if its root of a polynomial that...

a complex number z is said to be algebraic if its root of a polynomial that has inteher coefficients. Let A be the collection of algebraic numbers. Show that A is countable.

3. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf)...

3. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers. The greatest common divisor of a and b is written as gcd (a, b), or sometimes simply as (a, b). For example, gcd (12, 18) = 6, gcd (−4, 14) = 2 and gcd (5, 0) = 5. Two numbers are called co-prime or relatively prime...

I want a unique c++ code for the following. (Greatest Common Divisor) Given two integers x...

I want a unique c++ code for the following. (Greatest Common Divisor) Given two integers x and y, the following recursive definition determines the greatest common divisor of x and y, written gcd(x,y):     5 5 ± x y x y y x y y gcd( , ) if 0 gcd( , % ) if 0 Note: In this definition, % is the mod operator. Write a recursive function, gcd, that takes as parameters two integers and...

JAVA CLASS Greatest Common Divisor Finder Enter first number: 12 Enter second number: 8 Greatest common...

JAVA CLASS Greatest Common Divisor Finder Enter first number: 12 Enter second number: 8 Greatest common divisor: 4 Continue? (y/n): y Enter first number: 77 Enter second number: 33 Greatest common divisor: 11 Continue? (y/n): y Enter first number: 441 Enter second number: 252 Greatest common divisor: 63 Continue? (y/n): n The formula for finding the greatest common divisor of two positive integers x and y follows the Euclidean algorithm as follows: 1. Subtract x from y repeatedly until y...

Write the greatest common divisor over the given filed as a linear combination of the given...

Write the greatest common divisor over the given filed as a linear combination of the given polynomials. That is, given f(x) and g(x), find a(x) and b(x) so that d(x) = a(x)f(x) + b(x)g(x), where d(x) is the greatest common divisor of f(x) and g(x). (a) x^10 − x^7 − x^5 + x^3 + x^2 − 1 and x^8 − x^5 − x^3 + 1 over Q. (b) x^5 + x^4 + 2x^2 − x − 1 and x^3 +...

Let a and b be positive integers, and let d be their greatest common divisor. Prove...

Let a and b be positive integers, and let d be their greatest common divisor. Prove that there are infinitely many integers x and y such that ax+by = d. Next, given one particular solution x0 and y0 of this equation, show how to find all the solutions.

4. (a) Use the Euclidean algorithm to find the greatest common divisor of 21 and 13,...

4. (a) Use the Euclidean algorithm to find the greatest common divisor of 21 and 13, and the greatest common divisor of 34 and 21. (b) It turns out that 21 and 13 is the smallest pair of numbers for which the Euclidean algorithm requires 6 steps (for every other pair a and b requiring 6 or more steps a > 21 and b > 13). Given this, what can you say about 34 and 21? (c) Can you guess...

Activity 6. Write a Prolog definition of the greatest common divisor of two numbers. Then use...

Activity 6. Write a Prolog definition of the greatest common divisor of two numbers. Then use it to compute gcd(4, 10), gcd(15, 36), and gcd(25, 55). Activity 7. Write a Prolog program to find the last item in a list. give me the screenshot pls

If (x,y,z) is a primitive Pythagorean triple, prove that z= 4k+1

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