Question

Write a Pythagorean triplet whose smallest number is
(i). 6
(ii). 16

Answer 1

Hint: In this case, we are trying to find Pythagorean triplets with smallest numbers 6 and 16. Therefore, we should first understand what are Pythagorean triplets and thereafter we should take two standard Pythagorean triplets and then try to find out how we can generate the triplets with the smallest numbers given in the question.

Complete step-by-step answer:

A Pythagorean triplet (a,b,c) is defined as three numbers satisfying
${{a}^{2}}+{{b}^{2}}={{c}^{2}}.............(1.1)$
We notice that if (a,b,c) forms a Pythagorean triplet, then for any positive integer n,
\[\begin{align}
  & {{\left( na \right)}^{2}}+{{\left( nb \right)}^{2}}={{n}^{2}}\left( {{a}^{2}}+{{b}^{2}} \right) \\
 & ={{n}^{2}}{{c}^{2}}.............(1.2) \\
\end{align}\]
Where in the last line we have used equation (1.1). Thus, if (a,b,c) forms a Pythagorean triplet, then (na,nb,nc) also forms a Pythagorean triplet if n is a positive integer………………(1.3)
Also, as the same integer n is multiplied to all the numbers, the smallest number in (a,b,c) remains the smallest number in (na,nb,nc).
(i). We know that ${{3}^{2}}+{{4}^{2}}=9+16=25={{5}^{2}}$, therefore (3,4,5) forms a Pythagorean triplet. Therefore, taking n=2 in (1.3), we find that $\left( 2\times 3,2\times 4,2\times 5 \right)=(6,8,10)$ should also form a Pythagorean triplet and in (6,8,10), 6 is the smallest number. So, (6,8,10) is the answer to part (i) of the question.
(ii). We know that ${{8}^{2}}+{{15}^{2}}=64+225=289={{17}^{2}}$, therefore (8,15,17) forms a Pythagorean triplet. Therefore, taking n=2 in (1.3), we find that $\left( 2\times 8,2\times 15,2\times 17 \right)=(16,30,34)$ should also form a Pythagorean triplet and in (16,30,34), 16 is the smallest number. So, (16,30,34) is the answer to part (ii) of the question.
Thus, we obtain the answers to the given question as (6,8,10) for part (i) and (16,30,34) for part (ii).

Note: We should note that even though in the triplet (3,4,5), we can generate 16 by multiplying each element by 4 to get (12,16,20) and it will also be a valid Pythagorean triplet, we cannot have it as the answer to part (ii) because here the smallest number is 12 and not 16 and it has been asked to find the triplet where 16 should be the smallest number.