Question:

<p><strong>Step 1:</strong> Prime factorize 1008.</p><p>$1008 = 2^4 \times 3 \times 7$</p><p><strong>Step 2:</strong> Find even proper divisors. Even divisors must include at least one factor of 2.</p><p>Number of ways to select powers of 2 (at least 1): $4$ ways ($2^1, 2^2, 2^3, 2^4$)</p><p>Number of ways to select powers of 3: $2$ ways ($3^0, 3^1$)</p><p>Number of ways to select powers of 7: $2$ ways ($7^0, 7^1$)</p><p><strong>Step 3:</strong> Total even divisors = $4 \times 2 \times 2 = 16$</p><p><strong>Step 4:</strong> Exclude 1 and 1008 from proper divisors count. Since we already require at least one factor of 2, we only exclude 1008.</p><p>Required number of even proper divisors = $4 \times (2 + 1) \times (1 + 1) - 1 = 24 - 1 = 23$</p>