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....
Discrete Structures class. An ordered triple of positive integers (a,b,c) is called a Pythagorean Triple if...
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...
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.
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).
Matlab need it in keyboard types not handwriting 2) The Pythagorean theorem states that a2+b2-c2 . Write a MATLAB pro- gram in a script file that finds all the combinations of triples a, b, and c that are positive integers all smaller or equal to 50 that satisfy the Pythagorean the- orem. Display the results in a three-column table in which every row corre- sponds to one triple. The first three rows of the table are: 5 12 13 6...
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...
Co 2. (10 points) Let (a, b, c) be a primitive Pythagorean triple. Prove that cis of the form 4k1. a thec ( Rlats ie 2 2
4. Counting Problems. (a). Find number of ordered triples (x, y, z) of strictly positive integers such that 2 + y +z = 111. (b). Find the number of ways to arrange (all of) the letters of MATHEMATICS so that the result contains "EC" (so the letter E occurs immediately to the left of the letter C). (©). Find the number of ways to arrange (all of) the letters of MATHEMATICS so that the letter E occurs somewhere to the...
8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine all such triangles. b. Find the smallest two such triangles. 8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine...
Write a C program to find out the number of ordered pairs (a,b) of positive integers, such that: 1/a + 1/b = 3/2018.
DEFINITION: For a positive integer n, τ(n) is the number of positive divisors of n and σ(n) is the sum of those divisors. 4. The goal of this problem is to prove the inequality in part (b), that o(1)+(2)+...+on) < nº for each positive integer n. The first part is a stepping-stone for that. (a) (10 points.) Fix positive integers n and k with 1 <ksn. (i) For which integers i with 1 <i<n is k a term in the...