True or False—Every Baudhāyana triple is either a primitive triple or a scaled version of a primitive triple.

Question

Homi · Accepted Answer

We will show that any Baudhāyana triple must either be primitive or a scaled version of a primitive one.

Step 1 — What are Baudhāyana Triples?

A Baudhāyana triple is a set of three whole numbers (a, b, c). They follow the rule $a^2 + b^2 = c^2$ . For example, (3, 4, 5) is a Baudhāyana triple. Let us check: $3^2 + 4^2 = 9 + 16 = 25$ . And $5^2 = 25$ . So, $3^2 + 4^2 = 5^2$ . The Highest Common Factor (HCF) of 3, 4, and 5 is 1. When the HCF of a, b, and c is 1, we call it a primitive Baudhāyana triple.

Step 2 — What are Scaled Triples?

Consider another triple, like (6, 8, 10). Let us check if it is a Baudhāyana triple: $6^2 + 8^2 = 36 + 64$ $= 100$ $10^2 = 100$ So, $6^2 + 8^2 = 10^2$ . This means (6, 8, 10) is a Baudhāyana triple. Now, let us find the HCF of 6, 8, and 10. The common factors of 6, 8, and 10 are 1 and 2. The HCF of 6, 8, and 10 is 2. Since the HCF is not 1, (6, 8, 10) is not a primitive triple. What if we divide each number by the HCF, which is 2? We get $(6 \div 2, 8 \div 2, 10 \div 2)$ . This gives us the triple (3, 4, 5). We know (3, 4, 5) is a primitive Baudhāyana triple. So, (6, 8, 10) is just the primitive triple (3, 4, 5) multiplied by 2. We call (6, 8, 10) a scaled version of a primitive triple.

Step 3 — Proving the General Case

Let (a, b, c) be any Baudhāyana triple. This means $a^2 + b^2 = c^2$ . Let 'k' be the HCF of a, b, and c. This means we can write $a = kx$ , $b = ky$ , and $c = kz$ . Here, x, y, and z are whole numbers. The HCF of x, y, and z must be 1. If HCF(x, y, z) were greater than 1, then 'k' would not be the highest common factor of a, b, c. Now, let us substitute $a=kx$ , $b=ky$ , and $c=kz$ into the Baudhāyana rule: $(kx)^2 + (ky)^2 = (kz)^2$ $k^2x^2 + k^2y^2 = k^2z^2$ We can divide every term by $k^2$ (since k is a positive whole number). $\frac{k^2x^2}{k^2} + \frac{k^2y^2}{k^2} = \frac{k^2z^2}{k^2}$ $x^2 + y^2 = z^2$ This shows that (x, y, z) is also a Baudhāyana triple. Since HCF(x, y, z) = 1, (x, y, z) is a primitive Baudhāyana triple. And our original triple (a, b, c) is simply (kx, ky, kz). This means (a, b, c) is 'k' times the primitive triple (x, y, z). If k = 1, then (a, b, c) is itself a primitive triple. If k > 1, then (a, b, c) is a scaled version of a primitive triple. So, every Baudhāyana triple fits into one of these two categories.

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The statement is True.

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