GO Classes Weekly Quiz 1 | General Aptitude | Question: 9

Answer 1

We have proven the following proposition in the homework $2.$

Suppose $a,b,c\in \mathbb{Z}.$ If $a^2+b^2=c^2,$ then $a$ or $b$ is even.

An interesting consequence of this result is that in any right triangle in which all sides have integer length at least one of the two shorter sides must be of even length.

Suppose $a,b,c\in \mathbb{Z}. $ If $a^2+b^2=c^2,$ then $a$ or $b$ is even.

This result can be re-phrased as following :

$\textbf{Cool Application I:}$ Sums of odd perfect squares. Can a sum of two perfect squares be another perfect square? Sure; for example, $3^2+4^2=5^2, \; 5^2+12^2=13^2 \; 6^2+8^2=10^2, \; 7^2+24^2=25^2.$
However, no matter how much you try, you won't find any examples in which the two perfect squares on the left are both odd. Your task is to prove this, i.e.,

Prove that a sum of two odd perfect squares is never a perfect square.

(An interesting consequence of this result is that in any right triangle in which all sides have integer length at least one of the two shorter sides must be of even length.)

Comments

We have seen in “Proof Techniques – HW2 Solution – of 2024 Discrete Mathematics Course” that:

“In any right triangle whose all sides have integer length, at least one of the two shorter sides must be of even length.”

In the given right triangle, the two shorter sides are $a,b;$ So, at least one of them must be even.

It is NOT Necessary that both $a,b$ must be even.

The following three cases are possible:

• $a$ is even, $b$ is odd. (Example, $a = 4, b=3, c=5$)
• $b$ is even, $a$ is odd. (Example, $b = 4, a=3, c=5$)
• $a,b$ both are even. (Example, $a = 6, b=8, c=10$)

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