Primitive triples

The idea

A Baudhāyana triple with no common factor greater than 1 is called primitive; if all the primitive triples are found, all the Baudhāyana triples can be found.

Going the other way

Last time we multiplied a triple by k. Now go back: divide every number by a common factor.

Take (6, 8, 10). The number 2 divides all three numbers: 6, 8 and 10.

What do you think happens when we divide all three by 2?

• We get (3, 4, 5), still a triple
• It stops being a triple
• We get (12, 16, 20)

Divide it down

Each card is a Baudhāyana triple. The toy lists the common factors of its three numbers. Divide by a common factor, again and again, until only 1 is left.

Primitive triples

A Baudhāyana triple with no common factor greater than 1 is called primitive. (3, 4, 5) is primitive. (9, 12, 15) is not, because 3 divides all three numbers.

Every triple is a primitive triple scaled by some k. For example (12, 16, 20) = (3, 4, 5) × 4.

So if we can find all the primitive triples, we can find all Baudhāyana triples: just scale them.

Notes

A Baudhāyana triple with no common factor greater than 1 is primitive. (3, 4, 5) is primitive. (9, 12, 15) is not. Find all primitive triples and scaling gives all triples.

Check yourself

Which of these Baudhāyana triples is primitive?

The triple (15, 20, 25) is divided by the biggest common factor of its numbers. What is the new hypotenuse?

Answer: 5

5 divides 15, 20 and 25. We get (3, 4, 5), so the new hypotenuse is 25 ÷ 5 = 5.

(24, 45, 51) is a Baudhāyana triple. It is a primitive triple multiplied by k. What is k?

Answer: 3

3 divides 24, 45 and 51 and gives (8, 15, 17), which is primitive. So (24, 45, 51) = (8, 15, 17) × 3.

Why is it enough to find all the primitive triples?

• (6, 8, 10). 2 divides 6, 8 and 10. Dividing gives (3, 4, 5), so (6, 8, 10) is not primitive.
• (9, 12, 15). 3 divides 9, 12 and 15. Dividing gives (3, 4, 5), so (9, 12, 15) is not primitive.
• (5, 12, 13) — correct. Yes! The only number that divides 5, 12 and 13 is 1. It is primitive.
• Every triple is a primitive triple multiplied by some k, so scaling gives all the others. — correct. Yes! Find the primitive ones, and multiplying them by 1, 2, 3, ... gives every triple.
• Because primitive triples are the biggest ones.. They are the smallest in their family: (3, 4, 5) is smaller than (6, 8, 10) and (9, 12, 15).
• Because there are only a few triples.. There are infinitely many triples, as we saw by scaling.