document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210504>Question 1210504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If A and B are positive 2 digest integers, how many solutions are there of the equation 2a+3b=100 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853063 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3827)</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=853063\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red>Answer:  9</font>\r<BR>\n" );
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document.write( "Reasoning\r<BR>\n" );
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document.write( "It sounds like \"digest\" should be \"digit\".\r<BR>\n" );
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document.write( "I'm assuming the question is:<BR>\n" );
document.write( "\"If A and B are positive two-digit integers, how many solutions are there to the equation 2a+3b = 100?\"\r<BR>\n" );
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document.write( "Let's look at the related equation 2a+3b = 1, since we can multiply both sides by 100 later.\r<BR>\n" );
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document.write( "Through fairly quick trial-and-error, you should find an integer solution is (a,b) = (-1,1)<BR>\n" );
document.write( "Another method is to look at the graph of 2x+3y = 1 to find a lattice point, or you can use the Extended Euclidean Algorithm.\r<BR>\n" );
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document.write( "Since (-1,1) is a solution we can state that<BR>\n" );
document.write( "2a+3b = 1<BR>\n" );
document.write( "2*(-1)+3*(1) = 1\r<BR>\n" );
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document.write( "Multiply both sides by 100 to get,<BR>\n" );
document.write( "2*(-1)+3*(1) = 1<BR>\n" );
document.write( "100*(  2*(-1)+3*(1)  ) = 100*1<BR>\n" );
document.write( "100*2*(-1) + 100*3*(1) = 100<BR>\n" );
document.write( "2*( 100*(-1) ) + 3*(100*1) = 100<BR>\n" );
document.write( "2*( -100 ) + 3*(100) = 100<BR>\n" );
document.write( "This demonstrates that (a,b) = (-100,100) is an integer solution to 2a+3b = 100.<BR>\n" );
document.write( "Unfortunately a = -100 is not positive, and it's not a two-digit number either. <BR>\n" );
document.write( "A similar problem shows up with b as well.<BR>\n" );
document.write( "We must require that 10 <= a <= 99 and 10 <= b <= 99.<BR>\n" );
document.write( "The symbol <= means \"less than or equal to\".\r<BR>\n" );
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document.write( "2x+3y = 100 solves to y = (-2/3)x+(100/3)<BR>\n" );
document.write( "The slope -2/3 will tell us how to go from one integer point to another on this line.<BR>\n" );
document.write( "Start at (-100,100). <BR>\n" );
document.write( "Move down 2, right 3 to arrive at (-97,98)<BR>\n" );
document.write( "Move down 2, right 3 to arrive at (-94,96)<BR>\n" );
document.write( "etc\r<BR>\n" );
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document.write( "This \"down 2, right 3\" pattern can be expressed like this<BR>\n" );
document.write( "a = -100+3t<BR>\n" );
document.write( "b = 100-2t<BR>\n" );
document.write( "where t is an integer.<BR>\n" );
document.write( "If t = 0, then it leads to (a,b) = (-100,100)<BR>\n" );
document.write( "If t = 1, then it leads to (a,b) = (-97,98)<BR>\n" );
document.write( "If t = 2, then it leads to (a,b) = (-94,96)<BR>\n" );
document.write( "and so on.<BR>\n" );
document.write( "If t was negative, then you go the opposite direction \"up 2, left 3\".\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "10 <= a <= 99<BR>\n" );
document.write( "10 <= -100+3t <= 99<BR>\n" );
document.write( "10+100 <= -100+3t+100 <= 99+100<BR>\n" );
document.write( "110 <= 3t <= 199<BR>\n" );
document.write( "110/3 <= 3t/3 <= 199/3<BR>\n" );
document.write( "36.67 <= t <= 66.33 approximately<BR>\n" );
document.write( "Since t is an integer we round the lower bound up to the nearest integer; while the upper bound gets pulled down to the nearest integer.<BR>\n" );
document.write( "Note how t = 36 is too small since it leads to a = 8; while t = 67 is too large because it leads to a = 101.\r<BR>\n" );
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document.write( "Anyways, we go from <BR>\n" );
document.write( "36.67 <= t <= 66.33<BR>\n" );
document.write( "to<BR>\n" );
document.write( "37 <= t <= 66\r<BR>\n" );
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document.write( "Meanwhile,<BR>\n" );
document.write( "10 <= b <= 99<BR>\n" );
document.write( "10 <= 100-2t <= 99<BR>\n" );
document.write( "10-100 <= 100-2t-100 <= 99-100<BR>\n" );
document.write( "-90 <= -2t <= -1<BR>\n" );
document.write( "-90/(-2) >= -2t/(-2) >= -1/(-2) .... inequality signs flip<BR>\n" );
document.write( "45 >= t >= 0.5<BR>\n" );
document.write( "0.5 <= t <= 45<BR>\n" );
document.write( "1 <= t <= 45 .... rounding to nearest integer<BR>\n" );
document.write( "The inequality signs flipped because we divided all sides by a negative number.\r<BR>\n" );
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document.write( "Overlap these two intervals<BR>\n" );
document.write( "37 <= t <= 66<BR>\n" );
document.write( "1 <= t <= 45<BR>\n" );
document.write( "To get the interval 37 <= t <= 45<BR>\n" );
document.write( "I recommend drawing out a number line.\r<BR>\n" );
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document.write( "This set of t values will make both 10 <= a <= 99 and 10 <= b <= 99 true simultaneously where a = -100+3t and b = 100-2t.\r<BR>\n" );
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document.write( "How many integers are in the set {37,38,...,45}?<BR>\n" );
document.write( "We want the 37 to go to 1, the 38 to go to 2, etc.<BR>\n" );
document.write( "Subtract 36 from each item and you'll get the set {1,2,...,<font color=red>9</font>}\r<BR>\n" );
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document.write( "Or you could say 45-37+1 = <font color=red>9</font><BR>\n" );
document.write( "The formula I used is that there are n-m+1 items in the set {m,m+1,m+2,...,n-1,n} where m,n are integers and m < n.<BR>\n" );
document.write( "Note the -37+1 portion combines to -36, which is what we subtracted from each item of the previous set.\r<BR>\n" );
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document.write( "We determined there are <font color=red>9 integer solutions</font> to the equation 2a+3b = 100 where a,b are positive two-digit integers.<BR>\n" );
document.write( "Using a spreadsheet, or something like Python, the set of all solutions we're after is this<BR>\n" );
document.write( "a = 11,  b = 26<BR>\n" );
document.write( "a = 14,  b = 24<BR>\n" );
document.write( "a = 17,  b = 22<BR>\n" );
document.write( "a = 20,  b = 20<BR>\n" );
document.write( "a = 23,  b = 18<BR>\n" );
document.write( "a = 26,  b = 16<BR>\n" );
document.write( "a = 29,  b = 14<BR>\n" );
document.write( "a = 32,  b = 12<BR>\n" );
document.write( "a = 35,  b = 10<BR>\n" );
document.write( "Each time 'a' goes up by 3, b drops by 2.<BR>\n" );
document.write( "It seems like your teacher only cares about the number of solutions, rather than the solutions themselves. So this portion is likely optional.<BR>\n" );
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