<p><span style="color:#27ae60;"><span style="font-size:20px;">Solving the Problem</span></span></p>
<p>We know that the product of the two digits will equal 24. So, the two digits must be a factor pair of 24. We can use the <strong><span style="color:#8e44ad;">Nifty Factor Finding System</span></strong> to find all the factor pairs of single-digit integers:</p>
<ul>
<li>$24=1*24$</li>
<li>$24=2*12$</li>
<li>$24=3*8$</li>
<li>$24=4*6$</li>
</ul>
<p>We can see that the only single-digit pairs are $3,8$ and $4,6 $</p>
<p>Now we just have to figure out how many ways we can re-arrange these digit pairs to form 2-digit integers:</p>
<ul>
<li>$3,8$: $38$ and $83$</li>
<li>$4,6$: $46$ and $64$</li>
</ul>
<p>We can see that each pair gives us two integers, so in total we can form four two-digit integers.</p>
<p>So, the answer is <span style="color:#27ae60;">C</span>.</p>