Mathematics

Mathematics, 10.07.2019 04:50, pnhandley01

Prove that for all integers a and b, if a | b then a^2 | 3b^2.

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Mathematics, 25.06.2019 17:40, denvontgekingice

Suppose s is a set of 10 integers (any integers at all). prove that there are a pair of numbers in s whose difference is a multiple of 9; i.e. prove that there exists a, b ∈ s and an integer k ∈ z such that a − b = 9k. (hint: apply the pigeonhole principle to the potential remainders of elements of s with respect to division by 9).

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Mathematics, 05.07.2019 01:10, Syaralopez

5. (10 points) a pythagorean triple is a triple a, b, cof positive integers satisfying the pythagorcan theorem ab2 = 2. a pythagorean triple is called primitive if gcd(a, b, c) = 1 recall from assignment 1 that if a2 +b2 = 2, then a and b cannot both be odd. let a, b, c be a primitive pythagorean triple. then a and b cannot both be even, since otherwisc c is also even and gcd(a, b, c) 2. hence, exactly adopt the convention that a is odd and b is even (and thus c is odd) one of a and b is odd and the other is even. we (a) let s and t be odd integers such that s> t 2 1 and gcd(s, t) = 1. prove that b a= 2 2 forms a primitive pythagorean triple a,b, c (b) let and y be positive integers such that gcd(r, ) 1, and suppose that their product xy is a square. prove that both r and y are squares. (hint. usc the fundamental theorem of arithmetic.) (c) let a, b,c be a primitive pythagorean triple where a, b, c e z, a and c are even. prove that there exist odd integers s and t where s > t> 1and gcd(s, t) = 1 such that a,b,c and show that c+b and c-b are both squares and relatively prime.) (d) find all (not just primitive) pythagoran triples a, b,c of positive integers where odd, and bis 2-b2(cb) (c - b) given by the formulas above. (hint. write a2 are c c 15 =

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Mathematics, 21.08.2019 03:20, jasmine2919

[10 pts] prove that if 51 integers are selected from the first hundred positive integers, (1 100), there must be a pair of these integers with a sum equal to 101. 5.

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Mathematics, 22.08.2019 20:10, natishtaylor1p8dirz

Prove that if 51 integers are selected from the first hundred positive integers, (1 : s s 100), there must be a pair of these integers with a sum equal to 101.

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Prove that for all integers a and b, if a | b then a^2 | 3b^2....

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