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3. Determine the units in Z/nZ for n a non-negative integer. You can use without proving...
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer soluti...
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer solutions. After all this talk about trivial solutions, let's at least confirm that in some cascs non-trivial solutions do cxist. 9.15 Exercise. Find, by trial and error at least two non-trivial solutions to each of the Pell equations x2-2y2 I and x-3y21 Rolstcred by the cxistence of solutions for N 2 and N 3, our focus from this point...
10. A natural number n is called attainable if there exists non-negative integers a and b such th...
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
Let n be a non-negative integer. Letf() be such that f(x), f'(x).f"(x).,fn+exist, and are continu...
Let n be a non-negative integer. Letf() be such that f(x), f'(x).f"(x).,fn+exist, and are continuous, on an interval containing a. In this assignment, you will prove by induction on n that for any r in that interval f'(c) f"(c) fm (c) (t) (x -t)" dt. 7n n! 1. (a) Explain why the claim given above is true for n-0 (b) Use the fact that the claim is true for n-0 to explain why the claim is true for n =...
4. An element a in a ring R is called nilpotent if there exists a non-negative...
4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
Q3 3. How many solutions, in non-negative integers a, b and c, are there to the...
Q3
3. How many solutions, in non-negative integers a, b and c, are there to the linear equation a+b+c 73 with the restrictions a 2 30, b 30, and 10 7 Marks cs 30? Find the general solution of the recurrence: un-4un-13 x 4". 4. [8 Marks]
3. How many solutions, in non-negative integers a, b and c, are there to the linear equation a+b+c 73 with the restrictions a 2 30, b 30, and 10 7 Marks cs 30?...
(3) (a) A separated-variable solution to the heat equation is U(T) = cos(lx) cos(my) cos(nz)ept. What...
(3) (a) A separated-variable solution to the heat equation is U(T) = cos(lx) cos(my) cos(nz)ept. What is p in terms of the wavevector T = li + mj+nk? (b) In 1 space dimension the heat equation is aU/ət = kd’U/ax?. Try the non-separated- variable form: Utlob(t, x) = t-1/2e-z*/Axt. This represents a localized blob of heat spread- ing out. (c) In 2 space dimensions you can multiply the spreading blob solution in (b) with the same function Utlob(t, y) and...
C1= 5 C2= 6 C3= 10 GCD --> Greater Common Divisor B1 a. Let x :=...
C1= 5
C2= 6
C3= 10
GCD --> Greater Common Divisor
B1 a. Let x := 3C1 + 1 and let y := 5C2 + 1. Use the Euclidean algorithm to determine the GCD (x, y), and we denote this integer by g. b. Reverse the steps in this algorithm to find integers a and b with ax + by = g. c. Use this to find the inverse of x modulo y. If the inverse doesn't exist why not?...
We are given a system of two equations in the three unknowns x, y, and z....
We are given a system of two equations in the three unknowns x, y, and z. We transform an equation of the form ax + by + cz = d into the row [a, b, c, d]. We row reduce and find the matrix, Write down what the solutions are to this system given this information or explain why there is no solution. 1304 [0017] 0000
Please send me solutions for the above five questions. The questions are based on Pigeonhole Principle. 3. A shop co...
Please send me solutions for the above five
questions.
The questions are based on Pigeonhole Principle.
3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer...
Use matlab and write script: Please use n = input('Enter the value of n :') not...
Use matlab and write script: Please use n = input('Enter the value of n :') not the function command. Consider the following equation 6a + 9b + 20c = n The variables a, b, and c can be thought of as the number (non-negative integers) of 6’s, 9’s, and 20’s in n. Note that for a given n, there might be one, multiple, or no solutions for a, b, and c. For example, n = 15 → a = 1...
9.14 Theorem. f the natural mumber N is a perfect square, then the Pell equation Ny 1 has no non-trivial integer solutions. After all this talk about trivial solutions, let's at least confirm that in some cascs non-trivial solutions do cxist. 9.15 Exercise. Find, by trial and error at least two non-trivial solutions to each of the Pell equations x2-2y2 I and x-3y21 Rolstcred by the cxistence of solutions for N 2 and N 3, our focus from this point...
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
Let n be a non-negative integer. Letf() be such that f(x), f'(x).f"(x).,fn+exist, and are continuous, on an interval containing a. In this assignment, you will prove by induction on n that for any r in that interval f'(c) f"(c) fm (c) (t) (x -t)" dt. 7n n! 1. (a) Explain why the claim given above is true for n-0 (b) Use the fact that the claim is true for n-0 to explain why the claim is true for n =...
4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
Q3
3. How many solutions, in non-negative integers a, b and c, are there to the linear equation a+b+c 73 with the restrictions a 2 30, b 30, and 10 7 Marks cs 30? Find the general solution of the recurrence: un-4un-13 x 4". 4. [8 Marks]
3. How many solutions, in non-negative integers a, b and c, are there to the linear equation a+b+c 73 with the restrictions a 2 30, b 30, and 10 7 Marks cs 30?...
(3) (a) A separated-variable solution to the heat equation is U(T) = cos(lx) cos(my) cos(nz)ept. What is p in terms of the wavevector T = li + mj+nk? (b) In 1 space dimension the heat equation is aU/ət = kd’U/ax?. Try the non-separated- variable form: Utlob(t, x) = t-1/2e-z*/Axt. This represents a localized blob of heat spread- ing out. (c) In 2 space dimensions you can multiply the spreading blob solution in (b) with the same function Utlob(t, y) and...
C1= 5
C2= 6
C3= 10
GCD --> Greater Common Divisor
B1 a. Let x := 3C1 + 1 and let y := 5C2 + 1. Use the Euclidean algorithm to determine the GCD (x, y), and we denote this integer by g. b. Reverse the steps in this algorithm to find integers a and b with ax + by = g. c. Use this to find the inverse of x modulo y. If the inverse doesn't exist why not?...
Please send me solutions for the above five
questions.
The questions are based on Pigeonhole Principle.
3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer...
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