document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165989>Question 1165989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Starting with a positive integer, apply the following operations any number<BR>\n" );
document.write( "of times and in any order to produce a list of numbers:<BR>\n" );
document.write( "1. double the current term, or<BR>\n" );
document.write( "2. delete the last digit of the current term.<BR>\n" );
document.write( "An example of such a list is<BR>\n" );
document.write( "231, 23, 46, 92, 9, 18.<BR>\n" );
document.write( "(a) Find a list which starts with 51 and ends in 129.<BR>\n" );
document.write( "(b) Show that every starting number can produce a list ending in 1.<BR>\n" );
document.write( "A cycle is a list which eventually returns to the starting number, such as<BR>\n" );
document.write( "24, 48, 96, 9, 18, 36, 3, 6, 12, 24.<BR>\n" );
document.write( "(c) Show that every number from 1 to 41 occurs in a cycle with at most<BR>\n" );
document.write( "13 distinct terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852892 by </B><A HREF=/tutors/aboutme.mpl?userid=n2><B>n2(19)</B></A><img src=\"/images/stars/stars-06.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=n2><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=852892\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Starting with a positive integer, apply the following operations any number<BR>\n" );
document.write( "of times and in any order to produce a list of numbers:<BR>\n" );
document.write( "1. double the current term, or<BR>\n" );
document.write( "2. delete the last digit of the current term.<BR>\n" );
document.write( "An example of such a list is<BR>\n" );
document.write( "231, 23, 46, 92, 9, 18.<BR>\n" );
document.write( "(a) Find a list which starts with 51 and ends in 129.<BR>\n" );
document.write( "(b) Show that every starting number can produce a list ending in 1.<BR>\n" );
document.write( "A cycle is a list which eventually returns to the starting number, such as<BR>\n" );
document.write( "24, 48, 96, 9, 18, 36, 3, 6, 12, 24.<BR>\n" );
document.write( "(c) Show that every number from 1 to 41 occurs in a cycle with at most<BR>\n" );
document.write( "13 distinct terms.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this my post, &nbsp;I will answer question &nbsp;(a), &nbsp;ONLY.\r<BR>\n" );
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document.write( "To solve this problem, &nbsp;I asked &nbsp;Google &nbsp;AI &nbsp;OVERVIEW:\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "    Is it true that for every k-digit positive integer n, \r\n" );
document.write( "    there is a power of 2 whose first k digits coincide with the number n.\r\n" );
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document.write( "Below is the &nbsp;Google &nbsp;AI &nbsp;Overview answer.\r<BR>\n" );
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document.write( "The statement is true. [1]  \r\n" );
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document.write( "For every k-digit positive integer n, there is a power of two whose first digits coincide with the number n. \r\n" );
document.write( "This is a consequence of the fact that log_{10}(2) is an irrational number, which implies that the fractional parts \r\n" );
document.write( "of m*log_{10}(2) are uniformly distributed in the interval [0, 1) (by the Kronecker's Approximation Theorem). [1, 2, 3]  \r\n" );
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document.write( "Understanding the condition \r\n" );
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document.write( "For a positive integer P to be the leading digits of a power of two, say 2^m, it must satisfy the inequality: \r\n" );
document.write( "   P times 10^d <= 2^m < (P+1) times 10^d \r\n" );
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document.write( "for some integer d >= 0. This means 2^m has the same leading digits as P, followed by d more digits. \r\n" );
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document.write( "Answer: \r\n" );
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document.write( "The statement is true because any finite sequence of digits can be the leading digits of a power of two. \r\n" );
document.write( "This relies on the irrationality of log_{10}(2) and the uniform distribution of the fractional parts of its multiples. [1, 5]  \r\n" );
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document.write( "References\r\n" );
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document.write( "[1] https://www.quora.com/Prove-that-for-any-natural-number-n-there-esists-a-power-of-2-lets-say-2-k-such-that-2-k-in-its-decimal-expression-is-1-followed-by-n-zeros-and-eventually-other-digits-For-example-for-n-2-2-196-works-How-to-prove-it\r\n" );
document.write( "[2] https://www.antonellaperucca.net/didactics/Powers-of-2.pdf\r\n" );
document.write( "[3] https://math.stackexchange.com/questions/1370645/using-kroneckers-theorem-can-we-prove-theres-some-power-of-two-yielding-a-numb\r\n" );
document.write( "[4] https://math.stackexchange.com/questions/328655/proving-prime-p-divides-binompk-for-k-in-1-ldots-p-1\r\n" );
document.write( "[5] https://www.reddit.com/r/askmath/comments/18o86pr/is_it_true_that_for_any_positive_integer_n_there/\r\n" );
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document.write( "The link to this &nbsp;Google &nbsp;AI &nbsp;response is\r\n" );
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document.write( "https://www.google.com/search?q=Is+this+statement+true%3F+For+every+k-digit+positive+integer+n%2C+there+is+a+power+of+two+whose+first+n+digits+coincide+with+the+number+n.&rlz=1C1CHBF_enUS1071US1071&oq=Is+this+statement+true%3F+For+every+k-digit+positive+integer+n%2C+there+is+a+power+of+two++whose+first+n+digits+coincide+with+the+number+n.&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIHCAEQIRiPAjIHCAIQIRiPAjIHCAMQIRiPAtIBCTE3OTZqMGoxNagCCLACAfEFNet2ZYDKsLI&sourceid=chrome&ie=UTF-8\r\n" );
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document.write( "of Dec.4, 2025.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Then &nbsp;I &nbsp;asked &nbsp;Google &nbsp;AI &nbsp;another question:\r<BR>\n" );
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document.write( "    Find the degree of number 2 which starts 129.\r\n" );
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document.write( "Google &nbsp;AI &nbsp;answered:  &nbsp;the degree is &nbsp;110, &nbsp;and it is first degree of &nbsp;2 &nbsp;with this property.\r<BR>\n" );
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document.write( "Then I checked using &nbsp;MS &nbsp;Excel in my computer\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E110 ALIGN=MIDDLE  ALT=\"2%5E110\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.29807E+33.\r\n" );
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document.write( "Now I can present the desired sequence of operations\r<BR>\n" );
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document.write( "    51 -> 102 -> 10 -> 1 -> 2 -> 4 -> 8 -> 16 -> 32 -> 64 -> 128 -> . . . . -> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E110 ALIGN=MIDDLE  ALT=\"2%5E110\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 1.29807E+33,\r<BR>\n" );
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document.write( "which is a 34-digit integer number &nbsp;1298074214633706907132624082305024, &nbsp;whose three starting/leading digits are &nbsp;129.\r<BR>\n" );
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document.write( "Notice that I do not state that this sequence is the shortest possible.\r<BR>\n" );
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document.write( "I only state that this explicit sequence produces a desired number.<BR>\n" );
document.write( "</pre>\r<BR>\n" );
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document.write( "Thus part &nbsp;(a) &nbsp;is solved and the desired sequence of operations is presented explicitly.\r<BR>\n" );
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document.write( "Yes, &nbsp;in my solution &nbsp;I &nbsp;used help from &nbsp;Google &nbsp;AI &nbsp;Overview - I explicitly referred to it, <BR>\n" );
document.write( "so it is not stealing - it is normal work in contemporary environment. &nbsp;&nbsp;Thanks to &nbsp;Google &nbsp;AI &nbsp;for help &nbsp;(&nbsp;!&nbsp;)\r<BR>\n" );
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document.write( "And it suggests some art asking right questions and interpreting the &nbsp;AI &nbsp;answers properly,<BR>\n" );
document.write( "as well as organizing pieces and thoughts into a logically coherent text, <BR>\n" );
document.write( "which opens new knowledge and provides new material for teaching and learning.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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