Which statement is true about the Pythagorean triples 3-4-5 and 5-12-13?

Which statement is true about the Pythagorean triples 3-4-5 and 5-12-13?

• A. Only 3-4-5 is a Pythagorean triple
• B. Only 5-12-13 is a Pythagorean triple
• C. Neither
• D. Both 3-4-5 and 5-12-13 are Pythagorean triples

Answer: (D)

Pythagorean triples are sets of three integers a, b, c that satisfy a^2 + b^2 = c^2, with c the hypotenuse. For 3-4-5, 3^2 + 4^2 = 9 + 16 = 25, which is 5^2, so it fits the condition. For 5-12-13, 5^2 + 12^2 = 25 + 144 = 169, which equals 13^2, so this one also fits. Since both satisfy the Pythagorean relation, they are both Pythagorean triples. Both are primitive triples as well, because the three numbers in each set share no common divisor greater than 1. In general, if a^2 + b^2 = c^2 with integers, that triple holds, and any multiple of such a triple also works.

Pythagorean triples are sets of three integers a, b, c that satisfy a^2 + b^2 = c^2, with c the hypotenuse. For 3-4-5, 3^2 + 4^2 = 9 + 16 = 25, which is 5^2, so it fits the condition. For 5-12-13, 5^2 + 12^2 = 25 + 144 = 169, which equals 13^2, so this one also fits. Since both satisfy the Pythagorean relation, they are both Pythagorean triples. Both are primitive triples as well, because the three numbers in each set share no common divisor greater than 1. In general, if a^2 + b^2 = c^2 with integers, that triple holds, and any multiple of such a triple also works.