Another geometry problem? Noooo.

Prove that a right angled triangle with all integer sides has an area that is divisible by 6.


Let's look at the fundamental Pythagorean triplets instead of looking at their similar triangles. So we have the sides: (m2+n2),2mn,(m2n2)(m^2+n^2), 2mn, (m^2-n^2) , where m and n are two coprime natural numbers.

The area becomes: (mn)(m+n)mn.(m-n)(m+n)mn. As m and n are coprime, if they are of the same parity, then they both have to be odd and hence, a factor of 4 and hence 2, is established in side m2n2.m^2-n^2. If the parity is even for either one of them, then a factor of 2 is established in the product mn. Now we need to look for a factor of 3 in order to prove the question.

If either one of m and n is a multiple of 3, then there's no question about it since the factor gets established in the product mn.

Otherwise, W.L.O.G*, let m and n have the configuration:

  • 3k+1, 3l+1.
  • 3k+1, 3l+2.
  • 3k+2, 3k+2.


We can immediately see that in the first and last cases, the difference of m and n creates a factor of three in the area. In the second case, the sum creates a factor of three. Hence the area is going to be divisible by 6.


*= Without Loss Of Generality.

If you want to impress your friends and relatives at a dinner party or whatever, you can ask them to do this:

  • Think of any two numbers.
  • Add those two numbers, subtract those two numbers and multiply the two numbers.
  • Now write down your prediction on a piece of paper that says the product of the 3 resultant numbers is going to be a multiple of 3.
  • They will think that you got lucky. So let them have a go at it again and again and again.... till they give up and ask you how to do it. Explain the method and look like a genius.

Note by Vishnu C
6 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...