<PRE><B>Question 1172862</B>
&lt;br&gt;
The two given numbers are&lt;br&gt;
{{{12^24 = ((2^2)(3^1))^24 = (2^48)(3^24)}}}&lt;br&gt;
{{{16^18 = (2^4)^18 = 2^72}}}&lt;br&gt;
The LCM is&lt;br&gt;
{{{24^24 = ((2^3)(3^1))^24 = (2^72)(3^24)}}}&lt;br&gt;
So the given LCM is the LCM of just the two given numbers.&lt;br&gt;
That means the third number N can be any number of the form&lt;br&gt;
{{{((2^a)(3^b))}}}&lt;br&gt;
in which&lt;br&gt;
{{{0&lt;=a&lt;=72}}} (73 choices)&lt;br&gt;
and&lt;br&gt;
{{{0&lt;=b&lt;=24}}} (25 choices)&lt;br&gt;
ANSWER: The number of possible values of N is 73*25 = 1825.&lt;br&gt;
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