Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^...
Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^2 + b^2 = c^2. Examples are (3,4,5) and (5,12,13). Explain why for any Pythagorean triplet one must have that 12 divides abc. (Hint: It may be easiest to do this by showing that 3 divides abc and showing separately that 4 divides abc).
Homework Answers
Add Answer to:
Recall we say that a Pythagorean triple is a triplet of positive integers a, b and c such that a^...
Discrete Structures class. An ordered triple of positive integers (a,b,c) is called a Pythagorean Triple if...
Discrete Structures class.
An ordered triple of positive integers (a,b,c) is called a
Pythagorean Triple if a^2 + b^2 = c^2. Prove that if m & n are
pos int....
2) (10 pts) An ordered triple of positive integers (a, b, c) is called a Pythagorean Triple if a² + b2 = c2. Prove that if m and n are positive integers with m > n, then (m? – nº, 2mn, m² + n°) is a Pythagorean triple. Use this...
Re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode...
re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode (or Matlab code) that will output the number of unique Pythagorean triples that satisfy c< 200. [Unique means that you should not count (a 3, b 4, c 5) and (a 4, b 3, c 5) as two different triples, for example]. Explain why your program will work. Imagine a square n x n matrix A with diagonal elements which we believe to...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The prec...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c...
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c are natural numbers (in other words, they belong to N) and a2 +bc2. Prove that if a rectangle has sides of rational lengths p and q then the length of its diagonal is irrational if and only if there is a natural number r such that (p,q,r) is a a Pythagorean triple. State clearly any theorem proved in class that you use.
Given two integers a and b we say that a divides b, a | b, if...
Given two integers a and b we say that a divides b, a | b, if there is an integer k such that b = ka. Show "If a, b and c are integers such that a | b and a| c, then for all integers x,y we have to a (bx + cy)
(on this page, A, B, C, D are all positive integers and A/B <C/D.) We saw...
(on this page, A, B, C, D are all positive integers and A/B <C/D.) We saw in the previous assignment that CA CB - AD 1 DB=BD ? BD (The numerator must be an integer, and since the two fractions are unequal, it can't be 0.) In other words, "the closest two unequal rational numbers and can be is BD" (9.1) A sort of average of two fractions: . Show that <A+O- We gave an intuitive explanation of this in...
d) A Pythagorean quadruple consists of four integer a, b, c, d such that a2 +...
d) A Pythagorean quadruple consists of four integer a, b, c, d such that a2 + b2 c2d2 Turn some of the near-miss solutions that I talk about in the second part of the video into Pythagorean quadruples. (Hint: re- member that 1 12). e) Use some of the near-miss solutions to find some integers that can be written as the sum of two positive integer squares in two differ- ent ways.
Answer the following
Let \(\mathscr{F}\) be the set of Pythagorean triples in \(\mathbb{N}^{3}=\mathbb{N} \times \mathbb{N} \times \mathbb{N}\). That is, if \((a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) and \(a^{2}+b^{2}=c^{2}\) then \((a, b, c) \in \mathscr{T}\). Withoutloss of generality assume that \(a \leq b \leq c\). Let \((a, b, c)\) and \((e, f, g) \in \mathscr{F} .\) Define \((a, b, c) R(d, e, f) \Leftrightarrow \frac{d}{a}=\frac{e}{b}=\frac{f}{c}\).Note: If \((a, b, c) R(e, f, g)\) then \((a, b, c)\) and \((d, e, f)\) are...
1 For each of the following pairs of numbers a and b, calculate and find integers...
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
XVI. (Hint: See 8.1 and recall the work we did in class for this section.) a....
XVI. (Hint: See 8.1 and recall the work we did in class for this
section.)
a. Test the fairness of the die with α = 0.10
Numbers: 1 2 3 4 5 6
Obs: 43 58 42 39 57 61
b. Can you conclude that the die is fair? Why or why not.
XVII. (Hint: See 8.1 and recall the work we did in class for
this section.) From your text, page 420, do #4. Can you conclude
that the...
Discrete Structures class.
An ordered triple of positive integers (a,b,c) is called a
Pythagorean Triple if a^2 + b^2 = c^2. Prove that if m & n are
pos int....
2) (10 pts) An ordered triple of positive integers (a, b, c) is called a Pythagorean Triple if a² + b2 = c2. Prove that if m and n are positive integers with m > n, then (m? – nº, 2mn, m² + n°) is a Pythagorean triple. Use this...
re doing. A Pythagorean triple is a set of integers (a, b, c) satisfying a2+bc2. Write pseudocode (or Matlab code) that will output the number of unique Pythagorean triples that satisfy c< 200. [Unique means that you should not count (a 3, b 4, c 5) and (a 4, b 3, c 5) as two different triples, for example]. Explain why your program will work. Imagine a square n x n matrix A with diagonal elements which we believe to...
8.5 Theorem. Let s andt be any two different natural numbers with s t. Then (2st. (). is a Pythagorean triple. The preceding theorem lets us easily generale infinitely many Pythagorean triples, but, in fact, cvery primitive Pythagorean triple can be generated by chousing appropriale natural numbers s and and making the Pythagorean triple as described in the preceding thcorem. As a hint to the proof, we make a little observation. 8.6 Lemma. Let (a, b,e) be a primitive Pythagorean...
2. The triple (a, b, c) is called a Pythagorean triple if a, b and c are natural numbers (in other words, they belong to N) and a2 +bc2. Prove that if a rectangle has sides of rational lengths p and q then the length of its diagonal is irrational if and only if there is a natural number r such that (p,q,r) is a a Pythagorean triple. State clearly any theorem proved in class that you use.
(on this page, A, B, C, D are all positive integers and A/B <C/D.) We saw in the previous assignment that CA CB - AD 1 DB=BD ? BD (The numerator must be an integer, and since the two fractions are unequal, it can't be 0.) In other words, "the closest two unequal rational numbers and can be is BD" (9.1) A sort of average of two fractions: . Show that <A+O- We gave an intuitive explanation of this in...
d) A Pythagorean quadruple consists of four integer a, b, c, d such that a2 + b2 c2d2 Turn some of the near-miss solutions that I talk about in the second part of the video into Pythagorean quadruples. (Hint: re- member that 1 12). e) Use some of the near-miss solutions to find some integers that can be written as the sum of two positive integer squares in two differ- ent ways.
Let \(\mathscr{F}\) be the set of Pythagorean triples in \(\mathbb{N}^{3}=\mathbb{N} \times \mathbb{N} \times \mathbb{N}\). That is, if \((a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) and \(a^{2}+b^{2}=c^{2}\) then \((a, b, c) \in \mathscr{T}\). Withoutloss of generality assume that \(a \leq b \leq c\). Let \((a, b, c)\) and \((e, f, g) \in \mathscr{F} .\) Define \((a, b, c) R(d, e, f) \Leftrightarrow \frac{d}{a}=\frac{e}{b}=\frac{f}{c}\).Note: If \((a, b, c) R(e, f, g)\) then \((a, b, c)\) and \((d, e, f)\) are...
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
XVI. (Hint: See 8.1 and recall the work we did in class for this
section.)
a. Test the fairness of the die with α = 0.10
Numbers: 1 2 3 4 5 6
Obs: 43 58 42 39 57 61
b. Can you conclude that the die is fair? Why or why not.
XVII. (Hint: See 8.1 and recall the work we did in class for
this section.) From your text, page 420, do #4. Can you conclude
that the...
Most questions answered within 3 hours.
-
What is the significance of the constraints in a physical model?
What part do distinct constraints...
asked 1 hour ago -
a student was diagnosed with myopia (nearshightedness). He can
comfortably read a newspaper when it is...
asked 52 minutes ago -
Consider a 3m x 3m window in a house. The thermal conductivity
is reported to be...
asked 1 hour ago -
Why has California been the favorite destination of large number
of secondary migrants?
asked 2 hours ago -
Do not neglect the old for the new. The existing business must
not lose priority simply...
asked 5 hours ago -
Kylie is a single mom with two dependent children,
Tanner, age 7 and Olivia, age 11....
asked 6 hours ago -
Phosphorous + bromine = phosphorous tribromide. If 35.0 g of
bromine are reacted and 27.9 grams...
asked 7 hours ago -
Derive the long wavelength limit of the Planck energy density
distribution
asked 7 hours ago -
Calculate the pH of each of the following solutions.
0.50 M HBr
3.1×10−4 M KOH
4.2×10−5...
asked 11 hours ago -
For the year ended December 31, Depot Max’s cost of merchandise
sold was $85,600. Inventory at the...
asked 11 hours ago -
Week 10 - Professional Memo Assignment
Professional Memo Assignment
Your mission for this week, should you...
asked 11 hours ago -
Write a Python program that stores the data for each
player on the team, and it...
asked 11 hours ago