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To find the ordered triples of positive integers ( (A, B, C) ) such that ( ABC = 4000 ), we first factor 4000 into its prime factors: ( 4000 = 2^5 \times 5^3 ). The number of ways to distribute the prime factors among ( A ), ( B ), and ( C ) can be calculated using the stars and bars combinatorial method.
For the power of 2, we have ( 5 + 3 - 1 = 7 ) positions to place the dividers, yielding ( \binom{7}{2} = 21 ) ways. For the power of 5, we have ( 3 + 3 - 1 = 5 ) positions, yielding ( \binom{5}{2} = 10 ) ways. Multiplying these results gives ( 21 \times 10 = 210 ) ordered triples ( (A, B, C) ).
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What are Pythagoras' perfect numbers?
Pythagorean perfect numbers, also known as Pythagorean triples, are sets of three positive integers (a), (b), and (c) that satisfy the equation (a^2 + b^2 = c^2). A famous example is the triple (3, 4, 5), where (3^2 + 4^2 = 5^2). In number theory, perfect numbers are defined differently; they are positive integers that are equal to the sum of their proper divisors, like 6 or 28. However, Pythagorean perfect numbers specifically refer to the triples related to the Pythagorean theorem.
What are 10 pythagorean triples?
Pythagorean triples consist of three positive integers (a), (b), and (c) that satisfy the equation (a^2 + b^2 = c^2). Ten examples of Pythagorean triples include: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41), (12, 35, 37), (20, 21, 29), (12, 16, 20), (28, 45, 53), and (33, 56, 65). These sets represent the lengths of the sides of right triangles.
What isa pythagorean triples?
That refers to any solution of the equation a2 + b2 = c2, where a, b, and c are positive integers. For example, 3, 4, 5; or 5, 12, 13.
What are whole numbers that satisfy the pythagoreartheorm called?
Pythagorean Triples
How can you tell Pythagorean triples?
They are sets of three integers. The squares of two of them add up to the square of the third.
How many ordered triples of positive integers satisfy the equation a plus b plus c equals 6?
There are 6 such triples.
What are Pythagoras' perfect numbers?
Pythagorean perfect numbers, also known as Pythagorean triples, are sets of three positive integers (a), (b), and (c) that satisfy the equation (a^2 + b^2 = c^2). A famous example is the triple (3, 4, 5), where (3^2 + 4^2 = 5^2). In number theory, perfect numbers are defined differently; they are positive integers that are equal to the sum of their proper divisors, like 6 or 28. However, Pythagorean perfect numbers specifically refer to the triples related to the Pythagorean theorem.
How many ordered triples of three prime numbers exist for which the sum of the members of the triple is 24?
There are three triples making 15 ordered triples.There are three triples making 15 ordered triples.There are three triples making 15 ordered triples.There are three triples making 15 ordered triples.
Is Pythagoras' theorem the same as Pythagorean triples?
Pythagoras' theorem is a mathematical principle stating that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). Pythagorean triples are specific sets of three positive integers (a, b, c) that satisfy this theorem, such as (3, 4, 5) or (5, 12, 13). While the theorem describes the relationship between the sides of a right triangle, Pythagorean triples are concrete examples of integer solutions that adhere to this relationship.
What are 10 pythagorean triples?
Pythagorean triples consist of three positive integers (a), (b), and (c) that satisfy the equation (a^2 + b^2 = c^2). Ten examples of Pythagorean triples include: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41), (12, 35, 37), (20, 21, 29), (12, 16, 20), (28, 45, 53), and (33, 56, 65). These sets represent the lengths of the sides of right triangles.
What isa pythagorean triples?
That refers to any solution of the equation a2 + b2 = c2, where a, b, and c are positive integers. For example, 3, 4, 5; or 5, 12, 13.
What are whole numbers that satisfy the pythagoreartheorm called?
Pythagorean Triples
How do you find a pythagorean triple?
There are infinitely many Pythagorean triples. To find a Pythagorean triple take two positive integers x, y with x > y. A Pythagorean triple is of the form x2 - y2, 2xy, x2 + y2.
How can you tell Pythagorean triples?
They are sets of three integers. The squares of two of them add up to the square of the third.
What was Fermat's original proof of his last theorem?
Fermat's last theorem says there does not exist three positive integers a, b, and c which can satisfy the equation an + bn = cn for any integer value of n greater than 2. (2 with be pythagoran triples so we don't include that) Fermat proved the case for n=4, but did not leave a general proof. The proof of this theorem came in 1995. Taylor and Wiles proved it but the math they used was not even known when Fermat was alive so he could not have done a similar proof.
Which set of integers is a Pythagorean triple?
Pythagorean triples: 3, 4 and 5 or 5, 12 and 13 are two of them
How does Euclid's Formula Work?
Euclid's Formula is a method of generating Pythagorean Triples. A Pythagorean Triple is a set of three positive integers (whole numbers), which satisfy the equation a2 + b2 = c2. The smallest Pythagorean Triple is 3, 4, 5. Euclid's Formula says this: If you choose two positive integers m and n, with m < n, then the three numbers n2 - m2, 2mn and n2 + m2 form a Pythagorean Triple. For example, if m = 5 and n = 7, n2 - m2 = 49 - 25 = 24, 2mn = 70, and n2 + m2 = 49 + 25 = 74. 24, 70, 74 is a PT, because 242 + 702 = 742. That's how to use Euclid's Formula. If the question means why does it work, then: (n2 - m2)2 + (2mn)2 = (n4 + m4 - 2n2m2) + (4m2n2) = n4 + m4 + 2n2m2, which is the same thing as (n2 + m2)2 . Two things to note are: The Formula does not generate all possible Triples, and it will generate Primitive Triples (ones with no common factor), only if m and n have no common factor, (except 1).
