2
$\begingroup$

Given $x,y,z \in \mathbb{N},$ find probability that $x^2+y^2+z^2$ is divisible by $7.$

Attempt: from given condition

$x,y,z \in \{7k+1,7k+2,7k+3,7k+4,7k+5,7k+6,7k+7\}, k\in \mathbb{N}$

I'm not able to form different cases, help me to solve it, thanks.

  • 0
    What are squares congruent to mod 7? – 2017-02-27
  • 0
    This question is incomplete. What is the distribution of $x,y,z$? Are they picked uniformly? – 2017-02-27
  • 2
    @Durgesh Tiwari: To make the term "probability" meaningful for this problem, you could assume $x,y,z$ are chosen randomly and independently from the set $\{0,1,2,3,4,5,6\}$. – 2017-02-27
  • 4
    It's absolutely malicious to close this question because the assumed probability model is not explicitly declared. Occam's razor (https://en.wikipedia.org/wiki/Occam%27s_razor) tells us to assume that $x$, $y$, $z$ are independently uniformly distributed mod $7$. – 2017-02-27

2 Answers 2

3

Hint:

We know that for $k\in \mathbb N $, $$k^2 \equiv 0,1,2,4 \pmod 7$$

Hope you can take it from here.

2

The main part of solution:

a square of an integer can be only $0,1,2,4$ modulo 7 (it can be easily checked by taking squares of $7k+i$ for $i=0,1,2,3,4,5,6$ and investigating the remainder), so the sum of 3 squares is 0 mod 7 iff all of these numbers are divisible by 7 or their quadratic residues modulo 7 are $1,2,4$.