document.write( "Question 577782: In a three digit number , the hundreds digit , the tens digit and the units digit are in descending order and in Arithmetic progression . Each digit of the number is multiplied by the sum of the other 2 digits . The sum of all such results is A. B is the product of the tens digit and the sum of all the digits and A = (4/3)B . Find the number of such numbers ..\r
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\n" ); document.write( "\n" ); document.write( "a)4 b)3 c)2 d) More than 4 \n" ); document.write( "
Algebra.Com's Answer #370431 by Edwin McCravy(20056)\"\" \"About 
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In a three digit number , the hundreds digit , the tens digit and the units digit are in descending order and in Arithmetic progression . Each digit of the number is multiplied by the sum of the other 2 digits . The sum of all such results is A. B is the product of the tens digit and the sum of all the digits and A = (4/3)B . Find the number of such numbers ..
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document.write( "Let\r\n" );
document.write( "h = the hundredths digit\r\n" );
document.write( "t = the tens digit\r\n" );
document.write( "u = the units digit\r\n" );
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document.write( "Since the hundreds digit , the tens digit and the units digit are in \r\n" );
document.write( "descending order and in arithmetic progression, the units digit is\r\n" );
document.write( "the smallest.  Let d = the common difference in the arithmetic\r\n" );
document.write( "progression.  So that d will be positive, we take the units digit as\r\n" );
document.write( "the first term in the arithmetic progression, and the digits are then\r\n" );
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document.write( "u, t = u+d, h = u+2d\r\n" );
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document.write( "Each digit of the number is multiplied by the sum of the other 2 digits. \r\n" );
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document.write( "h(t+u) = (u+2d)(u+d + u) = (u+2d)(2u+d) = 2uČ+ud+4ud+2dČ = 2uČ+5ud+2dČ 
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document.write( "t(h+u) = (u+d)(u+2d + u) = (u+d)(2u+2d) = 2uČ+2ud+2ud+2dČ = 2uČ+4ud+2dČ
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document.write( "u(h+t) = u(u+2d + u+d) = u(2u+3d) = 2uČ+3ud
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\n" ); document.write( "The sum of all such results is A.
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document.write( "A = 2uČ+5ud+2dČ + 2uČ+4ud+2dČ + 2uČ+3ud = 6uČ+12ud+4dČ\r\n" );
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\n" ); document.write( "B is the product of the tens digit and the sum of all the digits
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document.write( "B = t(h+t+u) = (u+d)(u+2d + u+d + u) = (u+d)(3u+3d) = 3uČ+3ud+3ud+3dČ = 3uČ+6ud+3dČ \r\n" );
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document.write( "A = \"4%2F3\"B\r\n" );
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document.write( "6uČ+12ud+4dČ = \"4%2F3\"(3uČ+6ud+3dČ)\r\n" );
document.write( "6uČ+12ud+4dČ = 4uČ+8ud+4dČ\r\n" );
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document.write( "Here we have a polynomial equation in\r\n" );
document.write( "more than one variable in which all terms\r\n" );
document.write( "are of the same degree.  In such equations\r\n" );
document.write( "we must be careful. \r\n" );
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document.write( "2uČ-4ud = 0\r\n" );
document.write( "2u(u-d) = 0\r\n" );
document.write( " u(u-d) = 0\r\n" );
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document.write( "u = 0    u-d = 0\r\n" );
document.write( "           u = d\r\n" );
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document.write( "For the case u = 0, u is independent of d,\r\n" );
document.write( "and when we substitute u = 0 in\r\n" );
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document.write( "6uČ+12ud+4dČ = 4uČ+8ud+4dČ\r\n" );
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document.write( "we get identity 4dČ = 4dČ.  So we can\r\n" );
document.write( "proceed to get solutions.\r\n" );
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document.write( "t = u+d = 0+d = d \r\n" );
document.write( "h = u+2d = 0+2d = 2d\r\n" );
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document.write( "d   h=2d  t=d   u=0    number   \r\n" );
document.write( "1    2     1     0       210\r\n" );
document.write( "2    4     2     0       420\r\n" );
document.write( "3    6     3     0       630\r\n" );
document.write( "4    8     4     0       840\r\n" );
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document.write( "u = d   \r\n" );
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document.write( "Since u depends on the value of u, we\r\n" );
document.write( "must find what is allowed in order that\r\n" );
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document.write( "6uČ+12ud+4dČ = 4uČ+8ud+4dČ\r\n" );
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document.write( "be an identity\r\n" );
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document.write( "If u = d\r\n" );
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document.write( "6dČ+12(d)d+4dČ = 4dČ+8(d)d+4dČ\r\n" );
document.write( "  6dČ+12dČ+4dČ = 4dČ+8dČ+4dČ\r\n" );
document.write( "          22dČ = 16dČ\r\n" );
document.write( "           7dČ = 0\r\n" );
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document.write( "Which is true ONLY if d = 0\r\n" );
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document.write( "However if d = 0, and u = d, then \r\n" );
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document.write( "u and d are both 0, and\r\n" );
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document.write( "t = u+d = 0+0 = 0\r\n" );
document.write( "h = u+2d = 0+2(0) = 0\r\n" );
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document.write( "But 000 is not acceptable as a three\r\n" );
document.write( "digit number.\r\n" );
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document.write( "Therefore there are exactly 4 solutions.\r\n" );
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document.write( "Edwin
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