document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170970>Question 1170970</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the integers x and y. When y is divided by x the remainder is 29. When y is divided by x/2 the remainder is 13, Determine X </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795841 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795841\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x and y are integers<BR>\n" );
document.write( "Let's say they are also positive, so x > 0 and y > 0.<BR>\n" );
document.write( "Dividing y over x leads to some quotient, which we can call some integer q, and a remainder 29.<BR>\n" );
document.write( "This means, y/x = q + 29/x which is equivalent to y = qx + 29 after we clear out the fractions (multiply both sides by x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let p be another integer such that<BR>\n" );
document.write( "y/(x/2) = p + 13/(x/2)<BR>\n" );
document.write( "so we're dividing y over (x/2) and getting some quotient p and remainder 13.<BR>\n" );
document.write( "Multiplying both sides by the quantity (x/2) will clean things up a bit to get<BR>\n" );
document.write( "y = p(x/2) + 13\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiplying both sides by 2 gets us<BR>\n" );
document.write( "2y = px + 26<BR>\n" );
document.write( "The fractions are completely gone at this point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now plug in y = qx + 29 <BR>\n" );
document.write( "2y = px + 26<BR>\n" );
document.write( "2(qx+29) = px + 26<BR>\n" );
document.write( "2qx+58 = px+26\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's collect the x terms to one side<BR>\n" );
document.write( "Get everything else to the other side<BR>\n" );
document.write( "2qx+58 = px+26<BR>\n" );
document.write( "58-26 = px-2qx<BR>\n" );
document.write( "32 = px - 2qx<BR>\n" );
document.write( "px - 2qx = 32<BR>\n" );
document.write( "x(p - 2q) = 32\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The values p and q are integers. A linear combination of integers leads to another integer result.<BR>\n" );
document.write( "So if p and q are integers, then so is p-2q.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If p-2q wasn't an integer, then x(p-2q) wouldn't be an integer since x is already an integer.<BR>\n" );
document.write( "In other words, if p-2q wasn't an integer, then it wouldn't be possible to have x(p-2q) = 32.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because the two integers x and p-2q multiply to 32, we know that x and p-2q are factors of 32.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consequently, this means x is at most 32. If x were any larger than 32, then p-2q would have to be some positive number less than 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's list out the factors of 32:<BR>\n" );
document.write( "1,2,4,8,16,32\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The values pair up like so:<BR>\n" );
document.write( "1*32 = 32<BR>\n" );
document.write( "2*16 = 32<BR>\n" );
document.write( "4*8 = 32\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of the list of factors, we need to see which have the property that y/x has remainder 29, where x takes on one of the values from {1,2,4,8,32}\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It turns out that we can eliminate 1,2,4, and 8 from the list since they are too small. If x were say x = 4, then y/x = y/4 could only lead to remainders 0, 1, 2, or 3. The largest possible remainder is 1 less than the divisor. To generalize that, we can say the largest possible remainder of y/x is r = x-1. So that's why we can rule out 1,2,4 and 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The only thing left is 32. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let x = 32.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's also make y = 29 so that y/x has remainder 29. We could pick infinitely many other y values, but y = 29 is the smallest positive one we can pick.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y/x = 29/32 = 0 remainder 29<BR>\n" );
document.write( "Any time the numerator is smaller than the denominator, the remainder is automatically just the numerator. Picture having 29 cookies and 32 friends at a party. There isn't enough to have each friend have 1 full cookie, so there are 29 left over (remainder).<BR>\n" );
document.write( "The first condition has been met.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's see if y = 29 works with the other equation<BR>\n" );
document.write( "First plug in x = 32 and simplify<BR>\n" );
document.write( "y/(x/2) = y/(32/2) = y/16<BR>\n" );
document.write( "Then plug in y = 29<BR>\n" );
document.write( "y/16 = 29/16 = 1 remainder 13<BR>\n" );
document.write( "We have 29 cookies divided among 16 friends. Each person gets 1 full cookie, which leaves 29-16 = 13 left over. So this pair of x and y values satisfies the second condition as well.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: x = 32<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );