document.write( "Question 1191126: In what bases b, does (b+8) divide into (7b+8) with no remainder \n" ); document.write( "

Algebra.Com's Answer #822894 by greenestamps(13203) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "The statement of the problem is a bit awkward in using the term \"bases\". I interpret the problem to mean this:

\n" ); document.write( "For what values of b does (b+8) divide into (7b+8) with no remainder?

\n" ); document.write( "Going with that interpretation....

\n" ); document.write( "The requirement is that $\"%287b%2B8%29%2F%28b%2B8%29\"$ be an integer n.

\n" ); document.write( "Perform the \"division\" as quotient plus remainder:

\n" ); document.write( " $\"n=%287b%2B8%29%2F%28b%2B8%29\"$
\n" ); document.write( " $\"n=%287b%2B56-48%29%2F%28b%2B8%29\"$
\n" ); document.write( " $\"n=%28%287b%2B56%29-48%29%2F%28b%2B8%29\"$
\n" ); document.write( " $\"n=%287b%2B56%29%2F%28b%2B8%29-48%2F%28b%2B8%29\"$
\n" ); document.write( " $\"n=7-48%2F%28b%2B8%29\"$

\n" ); document.write( "In that last form of the equation, 7 is an integer, and we need n to be an integer; that means $\"48%2F%28b%2B8%29\"$ has to be an integer.

\n" ); document.write( "So (b+8) has to be a factor of 48:

\r\n" );
document.write( "        b+8   48  24  16  12   8   6   4   3   2   1\r\n" );
document.write( "          b   40  16   8   4   0  -2  -4  -5  -6  -7\r\n" );
document.write( "    48/(b+8)   1   2   3   4   6   8  12  16  24  48\r\n" );
document.write( "n=7-48/(b+8)   6   5   4   3   1  -1  -5  -9 -17 -41

\n" ); document.write( "The numbers in the second row of the table are all the integers for which (b+8) divides into (7b+8) with no remainder.

\n" ); document.write( "Examples....

\n" ); document.write( "b=16: b+8=24, 7b+8=112=8=120; 120/24=5

\n" ); document.write( "b=-4: b+8=4, 7b+8=-20; -20/4=-5

\n" ); document.write( "etc...

\n" ); document.write( " \n" ); document.write( "

\n" );