Question 1191126
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The statement of the problem is a bit awkward in using the term "bases".  I interpret the problem to mean this:<br>
For what values of b does (b+8) divide into (7b+8) with no remainder?<br>
Going with that interpretation....<br>
The requirement is that {{{(7b+8)/(b+8)}}} be an integer n.<br>
Perform the "division" as quotient plus remainder:<br>
{{{n=(7b+8)/(b+8)}}}
{{{n=(7b+56-48)/(b+8)}}}
{{{n=((7b+56)-48)/(b+8)}}}
{{{n=(7b+56)/(b+8)-48/(b+8)}}}
{{{n=7-48/(b+8)}}}<br>
In that last form of the equation, 7 is an integer, and we need n to be an integer; that means {{{48/(b+8)}}} has to be an integer.<br>
So (b+8) has to be a factor of 48:<br><pre>
        b+8   48  24  16  12   8   6   4   3   2   1
          b   40  16   8   4   0  -2  -4  -5  -6  -7
    48/(b+8)   1   2   3   4   6   8  12  16  24  48
n=7-48/(b+8)   6   5   4   3   1  -1  -5  -9 -17 -41</pre><br>
The numbers in the second row of the table are all the integers for which (b+8) divides into (7b+8) with no remainder.<br>
Examples....<br>
b=16: b+8=24, 7b+8=112=8=120; 120/24=5<br>
b=-4: b+8=4, 7b+8=-20; -20/4=-5<br>
etc...<br>