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In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area A are all positive integers. [1] [2] Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides 13, 14, 15 and area 84.
A least common multiple of a and b is a common multiple that is minimal, in the sense that for any other common multiple n of a and b, m divides n. In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. [10]
Lowest common factor may refer to the following mathematical terms: Greatest common divisor, also known as the greatest common factor; Least common multiple;
Euclid's formula for Pythagorean triples and the inverse relationship t = y / (x + 1) mean that, except for (−1, 0), a point (x, y) on the circle is rational if and only if the corresponding value of t is a rational number. Note that t = y / (x + 1) = b / (a + c) = n / m is also the tangent of half of the angle that is opposite the triangle ...
The ratio p/q takes its greatest value, 12/5=2.4, in Row 1 of the table, and is therefore always less than +, a condition which guarantees that p 2 − q 2 is the long leg and 2pq is the short leg of the triangle and which, in modern terms, implies that the angle opposite the leg of length p 2 − q 2 is less than 45°. The ratio is least in ...
m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor). lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and ...
An automedian triangle is one whose medians are in the same proportions (in the opposite order) as the sides. If x, y, and z are the three sides of a right triangle, sorted in increasing order by size, and if 2x < z, then z, x + y, and y − x are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5 ...
Every pair of congruence relations for an unknown integer x, of the form x ≡ k (mod a) and x ≡ m (mod b), has a solution (Chinese remainder theorem); in fact the solutions are described by a single congruence relation modulo ab. The least common multiple of a and b is equal to their product ab, i.e. lcm(a, b) = ab. [4]