Bunuel wrote:

If z is a positive integer and r is the remainder when z^2 + 2z + 4 is divided by 8, what is the value of r?

(1) When (z−3)^2 is divided by 8, the remainder is 4.

(2) When 2z is divided by 8, the remainder is 2.

A positive number \(x\) has the remainder \(\{0, \pm 1, \pm 2, \pm 3, \pm 4\}\) when divided by 8, hence \(x^2\) has the remainder \(\{0, 1, 4, 9, 16\}\) or \(\{0, 1, 4, 1, 0\}\) or \(\{0, 1, 4\}\) when divided by 8.

We have \(z^2+2z+4=(z+1)^2+3\).

Since \((z+1)^2\) has the remainder \(\{0, 1, 4\}\) when divided by 8,

\((z+1)^2+3\) has the remainder \(\{3, 4, 7\}\) when divided by 8, or r could be \(\{3, 4, 7\}\).

(1) If \((z-3)^2\) has the remainder 4 when divided by 8, then \(z-3\) has the remainder \(\pm 2\) or \(z\) has the remainder 1 or 5.

If \(z\) has the remainder 1, we have \(r=4\).

If \(z\) has the remainder 5, we have \(r=1\).

Insufficient.

(2) If \(2z\) has the remainder 2, or 10, then \(z\) has the remainder 1 or 5.

If \(z\) has the remainder 1, we have \(r=4\).

If \(z\) has the remainder 5, we have \(r=1\).

Insufficient.

Combine (1) and (2).

We still have 2 cases: \(z\) has the remainder 1 or 5. Insufficient. Answer D.

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