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Which of the following is a primitive Pythagorean triple?
- a)(6, 8, 10)
- b)(9, 12, 15)
- c)(5, 12, 13)
- d)(12, 16, 20)
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Which of the following is a primitive Pythagorean triple?a)(6, 8, 10)b...
Definition: A primitive Pythagorean triple consists of positive integers a, b, c with a2 + b2 = c2 and gcd(a,b,c) = 1.
Check (5, 12, 13): 52 + 122 = 25 + 144 = 169 = 132.
gcd(5,12,13) = 1, so the triple is primitive.
Therefore the correct answer is (5, 12, 13) (Option C). The other given triples are scalar multiples of (3,4,5) and hence not primitive.
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Which of the following is a primitive Pythagorean triple?a)(6, 8, 10)b...
Understanding Primitive Pythagorean Triples
A primitive Pythagorean triple consists of three positive integers (a, b, c) such that:
- They satisfy the Pythagorean theorem: a² + b² = c²
- The integers a, b, and c are coprime (their greatest common divisor is 1).
Evaluating the Options
Let's analyze each of the given options to identify the primitive Pythagorean triple.
Option A: (6, 8, 10)
- Check if they satisfy the Pythagorean theorem:
- 6² + 8² = 36 + 64 = 100 = 10²
- They have a common divisor of 2 (GCD is 2), so this is not primitive.
Option B: (9, 12, 15)
- Check Pythagorean theorem:
- 9² + 12² = 81 + 144 = 225 = 15²
- They have a common divisor of 3 (GCD is 3), so this is not primitive.
Option C: (5, 12, 13)
- Check Pythagorean theorem:
- 5² + 12² = 25 + 144 = 169 = 13²
- They have no common divisor other than 1 (GCD is 1), making this a primitive triple.
Option D: (12, 16, 20)
- Check Pythagorean theorem:
- 12² + 16² = 144 + 256 = 400 = 20²
- They have a common divisor of 4 (GCD is 4), so this is not primitive.
Conclusion
Among the options, only (5, 12, 13) meets both criteria of being a Pythagorean triple and being primitive. Therefore, option C is the correct answer.
A primitive Pythagorean triple consists of three positive integers (a, b, c) such that:
- They satisfy the Pythagorean theorem: a² + b² = c²
- The integers a, b, and c are coprime (their greatest common divisor is 1).
Evaluating the Options
Let's analyze each of the given options to identify the primitive Pythagorean triple.
Option A: (6, 8, 10)
- Check if they satisfy the Pythagorean theorem:
- 6² + 8² = 36 + 64 = 100 = 10²
- They have a common divisor of 2 (GCD is 2), so this is not primitive.
Option B: (9, 12, 15)
- Check Pythagorean theorem:
- 9² + 12² = 81 + 144 = 225 = 15²
- They have a common divisor of 3 (GCD is 3), so this is not primitive.
Option C: (5, 12, 13)
- Check Pythagorean theorem:
- 5² + 12² = 25 + 144 = 169 = 13²
- They have no common divisor other than 1 (GCD is 1), making this a primitive triple.
Option D: (12, 16, 20)
- Check Pythagorean theorem:
- 12² + 16² = 144 + 256 = 400 = 20²
- They have a common divisor of 4 (GCD is 4), so this is not primitive.
Conclusion
Among the options, only (5, 12, 13) meets both criteria of being a Pythagorean triple and being primitive. Therefore, option C is the correct answer.
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