Question

Another fiasco of wolframalpha:

How many integer solutions are there for the in-equation \[ x^2 +y^2 \le 25 \]?

And this is what wolframalpha says: http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2+%3C%3D+25

But I got a different answer, Just for fun, can you find the right answer?

Answer 1

@ash2326: I posted twice in the same group?

Answer 2

Obviously the maximum value \(x\) or \(y\) can take is \(5\), and the minimum is \(-5\). We can list all the choices as:
At \(x=\pm 5\), \(y\) can take only the value \(0\).
At \(x=\pm4\), \(y\) can take the values \(0, \pm1, \pm2\) and \( \pm3\).
At \(x=\pm 3\), \(y\) can take the values \(0, \pm1, \pm2, \pm3\) and \(\pm4\).
At \(x=\pm 2\), \(y\) can take the value \(0, \pm 1, \pm 2, \pm 3\) and \( \pm 4\).
At \(x=\pm1\), \(y\) can take the values \(0, \pm1, \pm 2, \pm 3\) and \(\pm 4\).
At \(x=0\) \(y\) can take the values \(0, \pm1, \pm2, \pm 3, \pm 4\) and \(\pm 5\).
So the total number of integer solutions is \(2+2(7)+2(9)+2(9)+2(9)+11=81.\)