True or False???

Number Theory Level 1

The difference between the squares of two consecutive odd integers is a multiple of 4. True or False?

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Rose Huang by Rose Huang , 19, China

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2 solutions

Roman Frago
Feb 13, 2015

( 2 x + 1 ) 2 − ( 2 x − 1 ) 2 = 4 x 2 + 4 x + 1 − 4 x 2 + 4 x − 1 = 8 x (2x+1)^{2} - (2x-1)^{2} \\= 4x^{2} +4x+1 -4x^{2} +4x -1 \\= 8x

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Hi I think it'd be better if you could copy and paste the LaTeX \LaTeX off my comment :

( 2 x + 1 ) 2 − ( 2 x − 1 ) 2 = 4 x 2 + 4 x + 1 − 4 x 2 + 4 x − 1 = 8 x (2x+1)^{2} - (2x-1)^{2} \\= 4x^{2} +4x+1 -4x^{2} +4x -1 \\= 8x

I think it'll look better this way ⌣ ¨ \ddot\smile

A Former Brilliant Member - 6 years, 3 months ago

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It is, but it is hard to type using a tablet...lol.

Roman Frago - 6 years, 3 months ago

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Yeah, I used to use an iPad before and I remember that I couldn't write solutions up due to the same difficulty .

A Former Brilliant Member - 6 years, 3 months ago
Lu Chee Ket
Feb 12, 2015

The difference is always 2 while sum of two odd numbers is always even number. Therefore, a contribution of 2 and 2 makes 4. Therefore, it is true.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

The sum of 2 consecutive odd numbers is already a multiple of 4 and the difference is always 2.

Roman Frago - 6 years, 4 months ago

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