document.write( "Question 1200551: In what bases, b, does (b + 6) divide into (5b + 6) without any remainder? \n" ); document.write( "

Algebra.Com's Answer #834709 by greenestamps(13215) $\"\"$ $\"About$

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\n" ); document.write( "The requirement is to find integers b for which (5b+6) divided by (b+6) is some integer k:

\n" ); document.write( " $\"%285b%2B6%29%2F%28b%2B6%29=k\"$
\n" ); document.write( " $\"k%28b%2B6%29=5b%2B6\"$
\n" ); document.write( " $\"bk%2B6k=5b%2B6\"$
\n" ); document.write( " $\"bk-5b=-6k%2B6\"$
\n" ); document.write( " $\"b%28k-5%29=-6k%2B6\"$
\n" ); document.write( " $\"b=%28-6k%2B6%29%2F%28k-5%29\"$

\n" ); document.write( "\"Perform the division\" like this:

\n" ); document.write( " $\"-6k%2B6=-6k%2B30-24=-6%28k-5%29-24\"$

\n" ); document.write( "So

\n" ); document.write( " $\"b=%28-6k%2B6%29%2F%28k-5%29=%28%28-6%28k-5%29-24%29%2F%28k-5%29%29\"$

\n" ); document.write( " $\"b=-6-24%2F%28k-5%29=-6%2B24%2F%285-k%29\"$

\n" ); document.write( "In that equation, -6 is an integer, and b must be an integer. That means $\"24%2F%285-k%29\"$ must be an integer; and since b must be positive, $\"24%2F%285-k%29\"$ must be greater than 6. Trying values of k that make $\"24%2F%285-k%29\"$ an integer greater than 6...

\n" ); document.write( "k=1; $\"24%2F%285-k%29=6\"$ ; b = -6+6 = 0 (not possible)

\n" ); document.write( "k=2; $\"24%2F%285-k%29=8\"$ ; b = -6+8 = 2

\n" ); document.write( "CHECK: (5b+6)/(b+6) = 16/8 = 2

\n" ); document.write( "k=3; $\"24%2F%285-k%29=12\"$ ; b = -6+12 = 6

\n" ); document.write( "CHECK: (5b+6)/(b+6) = 36/12 = 3

\n" ); document.write( "k=4; $\"24%2F%285-k%29=24\"$ ; b = -6+24 = 18

\n" ); document.write( "CHECK: (5b+6)/(b+6) = 96/24 = 4

\n" ); document.write( "ANSWERS: bases 2, 6, and 18

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