Question

Which one of the following is not a Pythagorean triples?

• A. $11,60,61$
• B. $16,63,65$
• C. $28,45,53$
• D. $30,80,89$

Answer 1

The correct option is D

$30,80,89$

Step 1. Explanation for the correct option:

Option (D): Find the numbers that do not form Pythagorean triples.

The Pythagorean triplet rule is $a^2+b^2=c^2$, where $a,b,\mathrm{and}c$ are positive integers.

Consider $30,80,89$.

$\begin{array}{rcl}{30}^2+{80}^2& =& {89}^2\\ & \Rightarrow & 900+6400\ne 7921\\ & \Rightarrow & 7300\ne 7921\end{array}$

This implies that $30,80,89$ do not form Pythagorean triples.

Hence, option (D) is correct.

Step 2. Explanation for the incorrect options:

Option (A): Find the numbers that do not form Pythagorean triples.

Consider $11,60,61$.

$\begin{array}{rcl}{11}^2+{60}^2& =& {61}^2\\ & \Rightarrow & 121+3600=3721\\ & \Rightarrow & 3721=3721\end{array}$

This implies that $11,60,61$ form Pythagorean triples.

Therefore, option (A) is incorrect.

Option (B): Find the numbers that do not form Pythagorean triples.

Consider $16,63,65$.

$\begin{array}{rcl}{16}^2+{63}^2& =& {65}^2\\ & \Rightarrow & 256+3969=4225\\ & \Rightarrow & 4225=4225\end{array}$

This implies that $16,63,65$ form Pythagorean triples.

Therefore, option (B) is incorrect.

Option (C): Find the numbers that do not form Pythagorean triples.

Consider $28,45,53$.

$\begin{array}{rcl}{28}^2+{45}^2& =& {53}^2\\ & \Rightarrow & 784+2025=2809\\ & \Rightarrow & 2809=2809\end{array}$

This implies that $28,45,53$ form Pythagorean triples.

Therefore, option (C) is incorrect.

Thus, options (A), (B), and (C) are incorrect.

Hence, option (D) is correct.