<PRE><B>Question 1097847</B>
&lt;br&gt;Learning how to solve this kind of problem using formal algebra is good training.  However, equally valuable is being able to use logical reasoning to solve this kind of problem.  Let&#39;s look at how we could solve this problem using logical analysis.&lt;br&gt;
&quot;A two-digit counting number has a value that is 8 times the sum of its digits.&quot;&lt;br&gt;
Okay; there is no obvious clue there, at least that I can see.... So&lt;br&gt;
&quot;6 times the units&#39; digit is 5 more than the tens&#39; digit&lt;br&gt;
Whoa!! If the units digit is very large, then 6 times it is going to be way more than 5 more than some single digit:
If the units digit is 1, then the tens digit would have to be 1 also (6*1 = 1+5).
If the units digit is 2, then the tens digit would have to be 7 (6*2 = 7+5).&lt;br&gt;
Clearly if the units digit is any larger, the &quot;tens digit&quot; could not be a single digit.&lt;br&gt;
So we only have two possibilities: the 2-digit number is either 11 or 72.&lt;br&gt;
Only 72 satisfies the first condition; so 72 is the number we are looking for.&lt;br&gt;
Studying mathematics is supposed to teach you to think; don&#39;t be afraid to exercise your brain by solving problems by thinking them through logically.</PRE>