document.write( "Question 1134705: In what bases, b, does (b+6) divide into (5b+6) without any remainder?\r
\n" ); document.write( "\n" ); document.write( "Note: I’m guessing that there are many bases that this could function because the question says in what BASES! \n" ); document.write( "

Algebra.Com's Answer #752201 by ikleyn(52866) $\"\"$ $\"About$

You can put this solution on YOUR website!
.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " The problem in the post is formulated by the very curved and twisted way, using outdated and antiquated language.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " In my opinion, the correct (adequate; straightforward and mathematically correct) formulation is AS FOLLOWS:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "            Find integer non-negative numbers \"b\" such that\r\n" );
document.write( "\r\n" );
document.write( "            (5b + 6) = 0  mod (b+6).\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " So, the problem is about solving congruences.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solution\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "    5b + 6 = 5(b+6) - 30 + 6,\r\n" );
document.write( "\r\n" );
document.write( "    5b + 6 = 5*(b+6) - 24.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore,\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    5b + 6 = 0  mod (b+6)      (1)\r\n" );
document.write( "\r\n" );
document.write( "if and only if\r\n" );
document.write( "\r\n" );
document.write( "    -24 = 0  mod (b+6),\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "which is equivalent to\r\n" );
document.write( "\r\n" );
document.write( "    24 = 0  mod (b+6).         (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From (2),  you easily find  b = 0, 2, 6, 18.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Check.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    a)  b = 0.  Then  5b + 6 = 5*0 + 6 = 0 + 6 = 6. From the other side,  b+6 = 6,  and  6 = 0 mod 6.   ! correct !\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    b)  b = 2.  Then  5b + 6 = 5*2 + 6 = 10 + 6 = 16. From the other side,  b+6 = 8,  and  16 = 0 mod 8.   ! correct !\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    c)  b = 6.  Then  5b + 6 = 5*6 + 6 = 30 + 6 = 36.  From the other side,  b+6 = 12,  and  36 = 0 mod 12.   ! Correct !\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    d)  b = 18.  Then  5b + 6 = 5*18 + 6 = 90 + 6 = 96.  From the other side,  b+6 = 24,  and  96 = 0 mod 24.   ! Correct !\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the problems has 4 solutions  b = 0, 2, 6 and 18.      ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "=================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The formulations and wordings as in the original post, are used now in sections and circles of amateurs of the OLD ENGLISH only.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In professional Math it is not in use just at least 200 years,
\n" ); document.write( "and in the School Math it is not in use more than 100 years.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is some \"before-Newton's era\" language . . . \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );