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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) ).
What else can I help you with?
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.
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.
What other things are related to the pythagorean theorem like the pythagoran triple?
Related to the Pythagorean theorem are Pythagorean triples, which are sets of three positive integers (a, b, c) that satisfy the equation (a^2 + b^2 = c^2). Additionally, the theorem is foundational in trigonometry, where it relates to the sine and cosine functions in right triangles. The concept of distance in the Cartesian coordinate system also derives from the Pythagorean theorem, as it calculates the distance between two points. Lastly, generalizations like the Law of Cosines extend these principles to non-right triangles.
How many ordered triples of positive integers satisfy the equation a plus b plus c equals 6?
There are 6 such triples.
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.
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).
What other things are related to the pythagorean theorem like the pythagoran triple?
Related to the Pythagorean theorem are Pythagorean triples, which are sets of three positive integers (a, b, c) that satisfy the equation (a^2 + b^2 = c^2). Additionally, the theorem is foundational in trigonometry, where it relates to the sine and cosine functions in right triangles. The concept of distance in the Cartesian coordinate system also derives from the Pythagorean theorem, as it calculates the distance between two points. Lastly, generalizations like the Law of Cosines extend these principles to non-right triangles.
Formula for midpoint of ordered triples?
You would use the midpoint formula on each axis, given that each ordered triple is represented by (x, y, z). The midpoint formula is another way of saying the mean of each axis.
What are the pythagorean theorem triples?
They are sets of integers such that the sum of the squares of two of the numbers equals the square of the third. For example, 5, 12 and 13 where 52 + 122 = 132
