Does this method yield non-primitive Baudhāyana triples? [Hint: Observe that among the triples generated, one of the smaller sidelengths is one less than the hypotenuse.]
No, this method does not yield non-primitive Baudhāyana triples.
Observation: In every triple generated by this method, one of the smaller sides is exactly one less than the hypotenuse (like 24 and 25, or 40 and 41).
Reasoning: Consecutive integers are always coprime (their greatest common divisor is 1).
∴ All triples generated by this method are primitive Baudhāyana triples.
Explanation
The chapter defines primitive Baudhāyana triples as those that do not have any common factor greater than 1. The hint points out that in triples generated by the odd square method, one smaller side is always one less than the hypotenuse. Two consecutive integers like n-1 and n are always coprime because any common divisor would have to divide their difference, which is 1. Therefore, the triple cannot have a common factor greater than 1, making all such triples primitive.
Solution Steps
Observe that in every triple generated, one smaller side is exactly one less than the hypotenuse.
Recognize that consecutive integers are always coprime (GCD = 1).
Conclude that all triples are primitive since they have no common factor greater than 1.