GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2
TRIPLES — Integer sides and primitive triples
Think first. Use AI purposefully. Verify always.
DISCOVER
Start with a sketch and a prediction.
Scale a 3–4–5 right triangle by 2. Does the equation survive? Does the new triple still have greatest common divisor 1?
Try on paper first. Use the model to test your idea before accepting the explanation.
Understand
A triple consists of positive integers a, b, c satisfying + = . It is primitive when gcd(a,b,c) = 1. Multiplication by a positive integer preserves the equation; scaling a primitive triple by k > 1 makes it non-primitive.
Key Ideas
(ka) + (kb) =
An integer right-triangle triple need not be primitive.
Visualise
Scale and test a triple
+ = . gcd = 1: primitive.
Method
Check positivity and integrality, put the largest number last, verify the squared equality, then compute the common divisor. Divide a non-primitive triple by its gcd to obtain a primitive one.
Worked Example
Given
The triple (15,36,39).
To Find
Its primitive triple and scale factor
Property
Divide every entry by their greatest common divisor.
Solution
- gcd(15,36,39) = 3.
- Dividing gives (5,12,13).
- + = and gcd(5,12,13) = 1.
Result
Primitive triple (5,12,13), scale factor 3.
AIMS Method
- A — Attempt independently
Sketch and predict before using a formula. - I — Investigate the idea
Separate given facts from visual assumptions. - M — Make connections
Connect area, length, algebra and geometric constructions. - S — Scrutinise your reasoning
Check each hypothesis and justify the conclusion.
Think first. Use AI purposefully. Verify always.
Focused Practice
List the primitive triples with every entry at most 20.
Check & Feedback
Name the known quantities and target, establish the theorem conditions and check your result in the original relationship.
Textbook connection: Ganita Prakash · Grade 8 · Part II · Chapter 2 — The Baudhāyana–Pythagoras Theorem · printed page 49