document.write( "Question 1099922: (1)-if 41x is a multiple of 11,where x is a digit what is the value of x
\n" ); document.write( "(2)-if 3y5 is a multiple of 11,where y is a digit what is the value of y
\n" ); document.write( "(3)-if 41z2 is a multiple of 6,where z is a digit what is the value of z \n" ); document.write( "

Algebra.Com's Answer #714637 by ikleyn(52778) $\"\"$ $\"About$

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document.write( "The divisibility by 11 rule says \r\n" );
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document.write( "    An integer number is divisible by  11  if and only if the alternate sum of its digits is divisible by  11.\r\n" );
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document.write( "    See my lesson  Divisibility by 11 rule  in this site.\r\n" );
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document.write( "So, in order for the 3-digit number 41x, where x is a missed digit (or a \"hidden\" digit),  was divisible by 11, the alternate sum \r\n" );
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document.write( "    4 - 1 + x\r\n" );
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document.write( "must be multiple of 11.\r\n" );
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document.write( "It is clear that we must (and can) to consider the only case when \r\n" );
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document.write( "   4 - 1 + x = 11,\r\n" );
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document.write( "which gives  x = 11 - 4 + 1 = 8.\r\n" );
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document.write( "Indeed, the number 418 is a multiple of 11:  418 = 11*38.\r\n" );
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\n" ); document.write( "\n" ); document.write( "(2) if 3y5 is a multiple of 11 , where y is a digit what is the value of y.\r
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document.write( "Solve it by the same way.\r\n" );
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document.write( "The equation  3 - y + 5 = 0  gives  y = 3 + 5 = 8.\r\n" );
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document.write( "Indeed, the number 385 is a multiple of 11:  385 = 11*35.\r\n" );
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document.write( "Here I used  0  as a unique appropriate multiple of 11.\r\n" );
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\n" ); document.write( "\n" ); document.write( "(3) if 41z2 is a multiple of 6, where z is a digit what is the value of z.\r
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document.write( "O-o-o, I finally got that z is the \"tens\" digit in this 4-digit number.\r\n" );
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document.write( "OK,  then we need to apply the divisibility rules for 3 and 2.\r\n" );
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document.write( "The \"divisibility by 2 rule\" is just satisfied, since the last digit of the number (\"ones\" digit) is even.\r\n" );
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document.write( "The \"divisibility by 3 rule\" requires the sum of the digits is multiple of 3:\r\n" );
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document.write( "    4 + 1 + z + 2  is divisible by 3\r\n" );
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document.write( "or, which is the same,  7+z must be divisible by 3.\r\n" );
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document.write( "So, the sum 7+z must be 9, or 12, or 15, which gives the possibilities for z to be equal 2, 5, 8.\r\n" );
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document.write( "Let us check three numbers  4122, 4152 and 4182 for divisibility by 6.\r\n" );
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document.write( "The answer is:  The numbers  4122,  4152 and  4182  all are multiples of 6.\r\n" );
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\n" ); document.write( "On divisibility rules for 2, 3 and 6 read in my lessons\r
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\n" ); document.write( "\n" ); document.write( "For other problems closely related to your in this post, see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Restore the omitted digit in a number in a way that the number is divisible by 9 and\r
\n" ); document.write( "\n" ); document.write( "    - Restore the omitted digit in a number in a way that the number is divisible by 11.\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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