document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150864>Question 1150864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good evening and Happy New Year to all Algebra Tutors!\r<BR>\n" );
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document.write( "What is the largest seven-digit number divisible by 44 that can be formed by the digits 1, 2, 3, 4, 6, 7, and 8 each used exactly once? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772409 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=772409\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Answer: <font color=red size=4>8761324</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "There are 7 digits, so there are 7! = 7*6*5*4*3*2*1 = 5040 different ways to arrange them. That's a lot of values to check. \r<BR>\n" );
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document.write( "We can make life easier by noticing that 44 = 4*11, so the answer must be divisible by 4 and 11. In other words, the answer is a multiple of 4 and a multiple of 11. \r<BR>\n" );
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document.write( "The divisibility by 4 rule is that the last two digits are divisible by 4. For example, the number 124 is divisible by 4 because 24 is too. Same for 1324 and 13524. We can have as many digits to the left of \"24\" and get the same result. This works because after reaching 100, the pattern for the multiples of 4 reset back to 00.\r<BR>\n" );
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document.write( "We want to pick the smallest two digit number that is a multiple of 4. Ideally each digit selected, of this two digit number, is as small as possible. The number 12 fits the description here. So we have some number that looks like this\r<BR>\n" );
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document.write( "__, __, __, __, __, 1, 2\r<BR>\n" );
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document.write( "The blank spaces will be filled with a single digit number from the set {3,4,6,7,8}\r<BR>\n" );
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document.write( "Let's try filling in those blank spaces such that we start with 8, then go to 7, and so on counting down til we get to 3. We skip 5 because its not part of the original set.\r<BR>\n" );
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document.write( "So we have<BR>\n" );
document.write( "__, __, __, __, __, 1, 2<BR>\n" );
document.write( "turn into<BR>\n" );
document.write( "8, 7, 6, 4, 3, 1, 2<BR>\n" );
document.write( "8764312\r<BR>\n" );
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document.write( "We already know it is divisible by 4 from the rule mentioned above, but as a check,<BR>\n" );
document.write( "8764312/4 = 2191078<BR>\n" );
document.write( "is a whole number so we confirm it is divisible by 4. From here on out, I'm only going to check for divisibility by 11 by using this division trick. We dont need to divide for divisibility of 4 checks because we're forcing the last two digits to be multiples of 4.\r<BR>\n" );
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document.write( "Check to see if the number is divisible by 11<BR>\n" );
document.write( "8764312/11 = 796755.636363637<BR>\n" );
document.write( "because we get a non-whole decimal number, we can see that 8764312 is not divisible by 11.\r<BR>\n" );
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document.write( "Let's try swapping the 3 and the 1<BR>\n" );
document.write( "we go from 8764<font size=4 color=blue>31</font>2 to 8764<font size=4 color=blue>13</font>2 (color used to show which digits swap)\r<BR>\n" );
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document.write( "Now check divisibility for 11<BR>\n" );
document.write( "8764132/11 = 796739.272727272 is not divisible by 11\r<BR>\n" );
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document.write( "Let's try having \"32\" as the last two digits. This is a multple of 4 (because 4*8 = 32). Now have 8 as the first digit, 7 as the second, and so on until you use up the digits. \r<BR>\n" );
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document.write( "We have this new number<BR>\n" );
document.write( "8764132\r<BR>\n" );
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document.write( "Check divisibility for 11<BR>\n" );
document.write( "8764132/11 = 796739.272727272 is not divisible by 11\r<BR>\n" );
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document.write( "Let's try swapping the 4 and the 1<BR>\n" );
document.write( "876<font size=4 color=blue>41</font>32 ---> 876<font size=4 color=blue>14</font>32\r<BR>\n" );
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document.write( "Check divisibility for 11<BR>\n" );
document.write( "8761432/11 = 796739.272727272 is not divisible by 11\r<BR>\n" );
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document.write( "Back to the drawing board. Let's try having \"24\" as the last two digits. Arrange the other digits such that we use the largest first, second largest next, and so on until the blank slots are filled. We have this number<BR>\n" );
document.write( "8763124\r<BR>\n" );
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document.write( "Check for divisibility of 11<BR>\n" );
document.write( "8763124/11 = 796647.636363637 doesnt work\r<BR>\n" );
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document.write( "Let's try swapping the 3 and the 1 again<BR>\n" );
document.write( "876<font size=4 color=blue>31</font>24 ---> 876<font size=4 color=blue>13</font>24\r<BR>\n" );
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document.write( "Check for divisibility of 4 and 11<BR>\n" );
document.write( "8761324/11 = 796484<BR>\n" );
document.write( "We finally get a whole number result. This means the number is a multiple of 11. \r<BR>\n" );
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document.write( "The value 8761324 is a multiple of 4 and a multiple of 11. So it is a multiple of 4*11 = 44. \r<BR>\n" );
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document.write( "As you can see this trial-and-error process is a bit lengthy and we got lucky in terms of finding a multiple of 44 that fit the description in the instructions. This is the largest such number because I forced the left most digits to be as large as possible. \r<BR>\n" );
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document.write( "There are other possible permutations of the digits {1,2,3,4,6,7,8} that are multiples of 44. If you are curious what they are, then check out the pastebin link below. I used pastebin so that I could save space on this current page to avoid clutter (the solution is already a bit too long as it is).\r<BR>\n" );
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document.write( "pastebin link<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://pastebin.com/qXYy9Skb\">https://pastebin.com/qXYy9Skb</a>\r<BR>\n" );
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document.write( "There is probably a much more efficient way to do this, but that is escaping me at the moment.<BR>\n" );
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