The Baudhayana-Pythagoras Theorem · Lesson 5 of 9
Right Triangles Having Integer Sidelengths
“Recognise integer triples, generate scaled versions, and distinguish primitive triples from their multiples.”
• Check whether three positive integers form a Baudhāyana–Pythagoras triple. • Find all triples whose entries are at most 20. • Explain why multiplying every entry by the same positive integer preserves a triple. • Identify primitive triples and reduce non-primitive triples. • Explain why infinitely many triples exist.
Many right triangles have a side length expressed by a square root. Yet a triangle with perpendicular sides 3 and 4 has a whole-number hypotenuse, 5. The same happens with 5, 12, and 13. These examples raise a new question: can we recognise and organise right triangles whose three side lengths are all positive integers, instead of discovering them one calculation at a time?
This question brings geometry and number patterns together. A triangle gives a relationship among lengths, while arithmetic lets you test that relationship and look for families of solutions. Some triples are enlarged copies of smaller ones; others give a new starting family.
We will first check examples directly, then explain a general scaling rule. Seeing several examples suggests a pattern, but an explanation is needed before you can rely on it for every positive integer.
What Is a Baudhāyana–Pythagoras Triple?
For the numbers 3, 4, and 5, calculate 3² + 4² = 9 + 16 = 25. The result equals 5². These positive integers therefore satisfy the right-triangle relationship. They form a Baudhāyana–Pythagoras triple.
The same triples are also called Baudhāyana triples, Pythagorean triples, or right-angled triangle triples. These names describe the same numerical condition. We usually list the two smaller entries first and the largest entry last, because the largest is the hypotenuse.
To verify a proposed triple, check two things: every entry must be a positive integer, and the sum of the squares of the two smaller entries must equal the square of the largest. Zero, negative lengths, and non-integer lengths do not belong to an integer triple as defined here.
A triple (a, b, c) of positive integers satisfying a² + b² = c², with c the largest entry. These integers can be the side lengths of a right triangle.
Problem
Determine whether (8, 15, 17) is a triple.
- 1.The entries are positive integers and the largest is 17.
- 2.Calculate 8² + 15² = 64 + 225 = 289.
- 3.Calculate 17² = 289.
- 4.The two results are equal, so (8, 15, 17) is a triple.
Problem
Determine whether (6, 8, 11) is a triple.
- 1.The largest entry is 11, so compare 6² + 8² with 11².
- 2.The smaller squares sum to 36 + 64 = 100.
- 3.But 11² = 121, which is different from 100.
- 4.The entries do not form a triple. Replacing 11 by 10 would make the equality work.
Finding Triples with Numbers up to 20
Organise the search instead of choosing numbers randomly. List the square numbers from 1² to 20². For each possible largest entry c, look for two smaller squares that add to c². To avoid duplicates, keep a less than b.
A useful equivalent search is to calculate c² − b² and check whether it is a positive perfect square a² with a < b. For example, when c = 13 and b = 12, the difference is 169 − 144 = 25 = 5², giving (5, 12, 13).
The complete list below keeps every entry at most 20. It includes four multiples of (3, 4, 5), but also two other triples. Multiples of one familiar triple are not the whole list.
| Triple | Sum of smaller squares | Largest square |
|---|---|---|
| (3, 4, 5) | 9 + 16 = 25 | 5² = 25 |
| (6, 8, 10) | 36 + 64 = 100 | 10² = 100 |
| (5, 12, 13) | 25 + 144 = 169 | 13² = 169 |
| (9, 12, 15) | 81 + 144 = 225 | 15² = 225 |
| (8, 15, 17) | 64 + 225 = 289 | 17² = 289 |
| (12, 16, 20) | 144 + 256 = 400 | 20² = 400 |
Create a table with columns c, b, c² − b², and a. Work through c from 5 to 20. You only need b values below c, and accept a result only when the difference is a positive perfect square with its square root smaller than b. Compare your successful rows with the six triples listed above. Explain how your ordering avoids counting (4, 3, 5) separately from (3, 4, 5).
Scaled Versions of a Triple
Compare (3, 4, 5), (6, 8, 10), and (9, 12, 15). The second triple multiplies every original entry by 2; the third multiplies every entry by 3. Geometrically, these are enlarged versions of the same right-triangle shape.
It is essential to multiply every entry by the same factor. Multiplying just the hypotenuse, or adding the same number to all entries, is a different operation and does not generally preserve the relationship.
For example, scaling (3, 4, 5) by 10 gives (30, 40, 50). We have 30² + 40² = 900 + 1600 = 2500 = 50². Scaling by 100 gives (300, 400, 500). Rather than checking forever, we can now explain why any positive integer factor works.
Why Scaling Always Works
Suppose a² + b² = c². Multiply each length by k. The square on the first new side has area (ka)² = k²a², and similarly the other new squares have areas k²b² and k²c².
The new sum is k²a² + k²b² = k²(a² + b²). Since the original bracket equals c², this becomes k²c² = (kc)². Thus (ka, kb, kc) is also a triple when k is a positive integer.
The geometric reason agrees with the algebra: multiplying every length by k multiplies every square area by k². The two sides of the area equality are multiplied by the same factor, so they remain equal.
Problem
Generate three scaled versions of (5, 12, 13).
- 1.With k = 2, multiply all entries by 2 to obtain (10, 24, 26).
- 2.With k = 3, the result is (15, 36, 39).
- 3.With k = 4, the result is (20, 48, 52).
- 4.For a direct check of the second result, 15² + 36² = 225 + 1296 = 1521 = 39².
Primitive Triples
The triple (9, 12, 15) has a common factor 3. Dividing all its entries by 3 gives the smaller triple (3, 4, 5). But the entries 3, 4, and 5 have no common factor greater than 1, so they cannot all be reduced further while staying integers.
A triple with no common factor greater than 1 is called primitive. The word refers to the three entries together; it does not mean that each entry is a prime number. For example, 4 is composite, yet (3, 4, 5) is primitive.
A Baudhāyana–Pythagoras triple whose three entries have greatest common factor 1.
Problem
Are (5, 12, 13) and (15, 36, 39) primitive?
- 1.First verify the relationship if needed: 25 + 144 = 169, and the second triple is a scale factor 3 version.
- 2.The numbers 5, 12, and 13 share no factor greater than 1, so the first triple is primitive.
- 3.The entries 15, 36, and 39 have greatest common factor 3, so the second is not primitive.
- 4.Dividing the second triple by 3 returns the first.
Reducing a Non-Primitive Triple
If f is a common factor of all three entries, divide a² + b² = c² by f². The resulting equation is (a/f)² + (b/f)² = (c/f)². Because f divides each entry, the new entries remain positive integers and form another triple.
To reach a primitive triple in one step, divide by the greatest common factor, not merely any common factor. Dividing (18, 24, 30) by 2 gives (9, 12, 15), which still has a common factor. Dividing by 6 gives (3, 4, 5), which is primitive.
Problem
Find the primitive triple underlying (24, 45, 51).
- 1.The common factor of all three entries is 3; no greater factor divides them all.
- 2.Divide each entry by 3 to obtain (8, 15, 17).
- 3.Check: 8² + 15² = 64 + 225 = 289 = 17².
- 4.The reduced entries share no common factor greater than 1, so this is the primitive form.
Why There Are Infinitely Many Triples
The triples (3k, 4k, 5k) are different for different positive integers k because their largest entries, 5k, are different. There is no largest positive integer, so this construction produces infinitely many triples.
This does not by itself prove that there are infinitely many primitive triples. The scaled versions with k greater than 1 share the factor k. In the next lesson, another pattern will generate new primitive triples instead of simply enlarging one existing triple.
Quiz
Which set is a Baudhāyana–Pythagoras triple?
What is obtained by scaling (8, 15, 17) by 3?
Which triple is primitive?
What is the primitive form of (18, 24, 30)?
Why does scaling by k preserve a triple?
Practice Problems
- Verify (12, 35, 37). Solution: 1. 12² + 35² = 144 + 1225 = 1369. 2. 37² = 1369, so it is a triple.
- Give five enlarged versions of (8, 15, 17). Solution: 1. Use factors 2, 3, 4, 5, and 6. 2. The triples are (16,30,34), (24,45,51), (32,60,68), (40,75,85), and (48,90,102). 3. Each is non-primitive because the scale factor divides all three entries.
- Reduce (30, 72, 78) to a primitive triple. Solution: 1. The greatest common factor is 6. 2. Divide to get (5,12,13). 3. Its common factor is 1, so the reduction is complete.
- Does adding 2 to every entry of (3,4,5) preserve a triple? Explain. Solution: 1. The resulting entries are (5,6,7). 2. 5² + 6² = 61, but 7² = 49. 3. Addition does not give a common multiplicative factor to the square areas, so the scaling proof does not apply.
- Explain why every triple is either primitive or a positive-integer multiple of a primitive triple. Solution: 1. Find the greatest common factor g of its entries. 2. If g = 1, the triple is already primitive. 3. If g > 1, divide every entry by g; division of the equation by g² preserves the triple condition. 4. The reduced entries have greatest common factor 1. Multiplying them by g recovers the original triple.
Key Takeaways
• A triple uses positive integers satisfying a² + b² = c². • Check the largest entry as the hypotenuse. • Multiplying every entry by a positive integer preserves the triple. • Square areas scale by the square of the length factor. • Primitive triples have greatest common factor 1; their entries need not all be prime. • Divide by the greatest common factor to recover a primitive triple. • Scaling a single triple already gives infinitely many triples.