Squares and Square Roots | IT

Question 26

Is $(5, 12, 13)$ a primitive Baudhayana triple? What are the other primitive Baudhayana triples with numbers less than or equal to $20$ ?

Solution

Understand the Question

A Baudhayana triple (or Pythagorean triple) is a set of three positive integers $(a, b, c)$ satisfying the relation $a^2 + b^2 = c^2$ .
A triple is primitive if $a$ , $b$ , and $c$ share no common factor other than $1$ , meaning $\text{HCF}(a, b, c) = 1$ .
Primitive triples are generated by the formulas $a = m^2 - n^2$ , $b = 2mn$ , and $c = m^2 + n^2$ , where $m > n > 0$ , $\text{HCF}(m, n) = 1$ , and one of $m, n$ is even while the other is odd.

(i) Is $(5, 12, 13)$ a primitive Baudhayana triple?

Step 1 · Verify the Baudhayana Relation

For $(5, 12, 13)$ , let $a = 5$ , $b = 12$ , and $c = 13$ .

Calculate $a^2 + b^2$ :

\begin{aligned} a^2 + b^2 &= 5^2 + 12^2 \\ &= 25 + 144 \\ &= 169 \end{aligned}

Calculate $c^2$ :

\begin{aligned} c^2 &= 13^2 \\ &= 169 \end{aligned}

Since $a^2 + b^2 = c^2 = 169$ , $(5, 12, 13)$ is a Baudhayana triple.

Step 2 · Check for Primitivity

Find the factors of each number:

Factors of $5$ : $1, 5$
Factors of $12$ : $1, 2, 3, 4, 6, 12$
Factors of $13$ : $1, 13$

The only common factor of $5, 12,$ and $13$ is $1$ , so $\text{HCF}(5, 12, 13) = 1$ .

Therefore, $(5, 12, 13)$ is a primitive Baudhayana triple.

Answer

(i) Yes, $(5, 12, 13)$ is a primitive Baudhayana triple.

(ii) What are the other primitive Baudhayana triples with numbers less than or equal to $20$ ?

Step 1 · Generate Primitive Triples with Values $\le 20$

Primitive Baudhayana triples $(a, b, c)$ can be generated using:

\begin{aligned} a &= m^2 - n^2 \\ b &= 2mn \\ c &= m^2 + n^2 \end{aligned}

where $m > n$ , $\text{HCF}(m, n) = 1$ , and one of $m, n$ is even while the other is odd.

Case 1: Let $m = 2, n = 1$

\begin{aligned} a &= m^2 - n^2 = 2^2 - 1^2 = 4 - 1 = 3 \\ b &= 2mn = 2 \times 2 \times 1 = 4 \\ c &= m^2 + n^2 = 2^2 + 1^2 = 4 + 1 = 5 \end{aligned}

Triple: $(3, 4, 5)$ (all values $\le 20$ ).

Case 2: Let $m = 3, n = 2$

\begin{aligned} a &= m^2 - n^2 = 3^2 - 2^2 = 9 - 4 = 5 \\ b &= 2mn = 2 \times 3 \times 2 = 12 \\ c &= m^2 + n^2 = 3^2 + 2^2 = 9 + 4 = 13 \end{aligned}

Triple: $(5, 12, 13)$ (the given triple).

Case 3: Let $m = 4, n = 1$

\begin{aligned} a &= m^2 - n^2 = 4^2 - 1^2 = 16 - 1 = 15 \\ b &= 2mn = 2 \times 4 \times 1 = 8 \\ c &= m^2 + n^2 = 4^2 + 1^2 = 16 + 1 = 17 \end{aligned}

Triple: $(15, 8, 17)$ or $(8, 15, 17)$ (all values $\le 20$ ).

Case 4: Let $m = 4, n = 3$

\begin{aligned} a &= m^2 - n^2 = 4^2 - 3^2 = 16 - 9 = 7 \\ b &= 2mn = 2 \times 4 \times 3 = 24 \\ c &= m^2 + n^2 = 4^2 + 3^2 = 16 + 9 = 25 \end{aligned}

Here $b = 24 > 20$ and $c = 25 > 20$ , so this triple is excluded.

For any $m \ge 5$ , $c = m^2 + n^2 \ge 5^2 + 1^2 = 26 > 20$ .

Excluding the given triple $(5, 12, 13)$ , the other primitive triples with numbers $\le 20$ are $(3, 4, 5)$ and $(8, 15, 17)$ .

Answer

(ii) $(3, 4, 5)$ and $(8, 15, 17)$

Common Mistakes

Primitive vs. Non-primitive: Mistaking multiples like $(6, 8, 10)$ or $(9, 12, 15)$ for primitive triples. Although they satisfy $a^2 + b^2 = c^2$ , they are not primitive because $\text{HCF}(a, b, c) > 1$ .
Including the Given Triple: Listing $(5, 12, 13)$ again in part (ii) despite the question asking for the other primitive triples.
Exceeding the Bound: Forgetting the condition that all three numbers must be $\le 20$ , which excludes triples like $(7, 24, 25)$ .

More questions in IT

How can one construct a square having double the area of a given square?

Why does the new dotted square have double the area of the original square?

In many of the constructions in the Śulba-Sūtra, it is desirable to construct, where needed, what Baudhāyana calls ‘east-west’ and ‘north-south’ lines, i.e., horizontal and vertical lines that are perpendicular to each other. Can you draw some horizontal and vertical lines to see why the new square has double the area of the original square? You could draw some horizontal and vertical lines as shown on the right.

Why should the extension of the vertical and horizontal sides of the original square pass through the vertices of the dotted square?

[Hint: From the diagonal property of a square, the line that bisects an angle passes through the opposite vertex. Argue why the vertical and horizontal sides of the original square bisect the two angles of the dotted square.**]

Context: So, the new square has double the area of the original square, because the original square is made up of two small triangles, while the new square is made up of four small triangles.

Q. Moreover, all these small triangles are congruent to each other. Can you explain why?

Now suppose we are given a square, and we want to construct a square whose area is half that of the original square. How would you do it?

Why is the smaller inside square half the area of the larger square?

Again, adding some east-west and north-south lines can explain it:

Why is $PQRS$ a square? Why is its area half that of the original paper?

Explain by connecting $QS$ and $PR$ , finding the different angles formed, and then using tringle congruence.

Find the hypotenuse of this isosceles right triangle.

What is the value of $\sqrt{2}$ ?

Q10

Is $\sqrt{2}$ less than or greater than 1?

Q11

Is $\sqrt{2}$ less than or greater than 2?

Q12

Can we find closer bounds for $\sqrt{2}$ ?

Q13

Will we ever get a number with a terminating decimal representation whose square is 2?

If there is such a terminating decimal starting with $1.414\dots$ whose square is 2, then it must have a non-zero last digit. If this is the case, then the decimal representation of its square will also have a non-zero last digit after the decimal point. For example, if $\sqrt{2}$ is of the form $1.414\dots4$ , then its square will be of the form—

Q14

Use this formula to check your answers in the Figure it Out on page 39.

Q15

What if we wish to combine two squares of 'different' sizes to make a large square whose area is the sum of the two smaller squares?

Q16

Why does Baudhāyana’s method work?

Q17

Can you see why the method works in the case where the two squares are the same size? Does it agree with the method we used earlier to combine two same sized squares into a bigger square?

Q19

Explain why all the angles of this new 4-sided figure are right angles and so it is a square.

Q20

List down all the Baudhāyana triples with numbers less than or equal to 20.

Q21

Is there an unending sequence of Baudhāyana triples?

Q22

Is $(30, 40, 50)$ a Baudhāyana triple?

Is $(300, 400, 500)$ a Baudhāyana triple?

Q23

Context: The list of Baudhāyana triples having numbers less than or equal to 20 contains the following triples — $(3, 4, 5)$ , $(6, 8, 10)$ , $(9, 12, 15)$ , $(12, 16, 20)$ .

Q. Do you see any pattern among them?

Q24

Context: All these triples can be obtained by multiplying each term of $(3, 4, 5)$ by a certain positive integer.

Q. Can we form a conjecture on Baudhāyana triples based on this observation?

Q25

Context: Conjecture: $(3k, 4k, 5k)$ is a Baudhāyana triple, where $k$ is any positive integer.

Q. Is this true?

Q26

Is $(5, 12, 13)$ a primitive Baudhayana triple? What are the other primitive Baudhayana triples with numbers less than or equal to $20$ ?

Q27

Generate 5 scaled versions of each of these primitive triples. Are these scaled versions primitive?

Q28

If $(a, b, c)$ is non-primitive, and the integers have $f$ — greater than 1 — as a common factor, then is $\left(\dfrac{a}{f}, \dfrac{b}{f}, \dfrac{c}{f}\right)$ a Baudhayana triple? Check this statement for $(9, 12, 15)$ . Justify this statement.