Prove them! (part 1)

Prove them if you can!

Whatever it's easy or not, I hope that you can try to prove the problems below. Sadly, I can't prove them... Try your best and please post your proof below and state what method/theorem etc. did you use.

1) Prove that the sum of two odd square numbers is always not divisible by 4.

2) Prove that any triangle can be cut into six similar triangles. (Your proof must be valid for all triangles and not just a specific type. )

Don't forget to write the question number when posting your proof.There are more problems! Wait for the next note!. Thanks for proving...!

#Geometry #NumberTheory #SimilarFigures #Proofs #Sumofsquares

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

First I will write a proof of the first, its easy though. Let the two odd square numbers be $(2n+1)^2$ and $(2k+1)^2$ . Now, on adding them, we get $(2n+1)^2+(2k+1)^2=4n^2+4n+1+4k^2+4k+1$ Which is $4(n^2+k^2+n+k)+2$ , which clearly leaves a remainder of 2 when divided by 4.

Dinesh Chavan - 6 years, 12 months ago

Thanks!

A Former Brilliant Member - 6 years, 12 months ago

How about the first question? Anyone proved it?

A Former Brilliant Member - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$