<p>If <i>r</i>, <i>s</i>, <i>t</i> are prime numbers and <i>p</i>, <i>q</i> are the positive integers such that their LCM of <i>p</i>, <i>q</i> is \(r^2t^4s^2\), then the numbers of ordered pair of (<i>p</i>, <i>q</i>) is</p>
Step-by-Step Solution
Key Concept: For LCM condition, the maximum exponent of each prime in the two numbers must equal the exponent in the LCM. Count valid exponent pairs carefully.
<p><strong>Solution:</strong> Let \(p = r^{a_1}s^{b_1}t^{c_1}\) and \(q = r^{a_2}s^{b_2}t^{c_2}\) where \(a_i, b_i, c_i \geq 0\).</p><p>Since \(\text{LCM}(p,q) = r^2s^2t^4\), we need \(\max(a_1, a_2) = 2\), \(\max(b_1, b_2) = 2\), \(\max(c_1, c_2) = 4\).</p><p>For each prime power: the number of ways to choose exponent pairs such that the maximum is the required value.</p><p>For \(\max(a_1, a_2) = 2\): pairs are \((0,2), (1,2), (2,0), (2,1), (2,2)\) = 5 ways</p><p>For \(\max(b_1, b_2) = 2\): 5 ways</p><p>For \(\max(c_1, c_2) = 4\): pairs are \((0,4), (1,4), (2,4), (3,4), (4,0), (4,1), (4,2), (4,3), (4,4)\) = 9 ways</p><p>Total ordered pairs = \(5 \times 5 \times 9 = 225\). However, we must exclude cases where both \(p=1\) and \(q=1\) simultaneously, giving \(225 - 1 = 224\).</p><p>∴ Answer is D.</p>
Correct Answer: D