Is this obvious?

Number Theory Level 2

Every perfect square number is one of the forms $5n$ , $5n \pm 1$ where $n$ is an integer.

Correct Incorrect

1 solution

Brian Charlesworth
Nov 22, 2015

Modulo $5,$ any integer $n$ is equivalent to one of $0, \pm 1$ or $\pm 2.$

Thus $n^{2}$ will be respectively equivalent to $0, 1$ or $4 \equiv -1$ modulo $5,$

i.e., the statement that every perfect square will be of one the forms $5n, 5n \pm 1$ is $\boxed{correct}.$

5 x 11 +1= 56; 5 x 11 -1= 54, both are not perfect squares.

Rezaul Kabir - 5 years, 6 months ago

Every perfect square is equivalent to 0,1,4 modulo 5. But a number being 0,1,4 modulo 5 doesn't guarantee it being a square.

A Former Brilliant Member - 5 years, 6 months ago

Please explain which integer makes that sentence right with 100. That is a perfect square

Monica Arboleda - 5 years, 6 months ago

@Monica Arboleda – I didn't get you. What sentence are you talking about? "Every perfect square is 0,1,4 modulo 5" ?

A Former Brilliant Member - 5 years, 6 months ago

@A Former Brilliant Member – Sorry. I had not seen the part 5n, before 5n+-1 never mind :(

Monica Arboleda - 5 years, 6 months ago