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 smallest integer whose first n multiples all contain a 3
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Monthly Archives: January 2016
Triangles with 60° angle and sides integers
Previous blog: Integer triangles with 120° angle Law of cosines https://en.wikipedia.org/wiki/Law_of_cosines If —> then, To find triples that satisfy the relation We use the parametric equations where … Continue reading
Integer Triangles such the median & Altitude to/from the largest side are all integers
To find triangles such that its sides, the median ( ) to the largest side and the altitude from are all integers. In triangle, the lengths … Continue reading
Integer Triangles (a,b,c) such that the median to the largest side is an integer
To find integer triangles such that the median ( ) to the largest side is an integer. In triangle, the lengths of the medians : For example, … Continue reading
Some solutions to Project Euler 62: Cubic permutations
Project Euler 62 https://projecteuler.net/index.php?section=problems&id=62 The cube, , can be permuted to produce two other cubes: and In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also … Continue reading
Integer triangles with 120° angle
Law of cosines https://en.wikipedia.org/wiki/Law_of_cosines If —> then, To find triples that satisfy the relation We use the parametric equations : where and are integers and … Continue reading
Triangle (A,B,C) such that tan A, tan B, and tan C integers?
For any triangle ABC, the proof can be easily derived from simple trigonometric identities. Is it possible for all three of , , and to be integers? If so, how … Continue reading
(x,y,z) positive integers, √x + √y = √z, z ≤ 1000
, and are positive integers such that Find all possible values of A pattern will emerge. Find it. Solution: must be a square. In general, , … Continue reading
Primitive Heron triangles – two sides have the common factor 5
The primitive Heron triangle (45, 296, 325) in which two sides have the common factor 5. area = 5328 perimeter = 666 Note that 325 is precisely the length of the … Continue reading
Heron Triangle with square sides
Heron Triangle with square sides Area = 32918611718880 Perimeter = 42216338 has 6 representations as a sum of 2 squares: … Continue reading
Heron triangles (17, n, n+3), (17, n, n+5) and (17, n, n+1)
Older post: Heron triangles (17, n, n+3) and (17, n, n+5) Here are the first few examples: