document.write( "Question 714806: Find a natural number for which if we move the first digit to the last place, the number that will emerge will be half of the original.\r
\n" ); document.write( "\n" ); document.write( "ex 321 goes to 213 but no a sollution
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Algebra.Com's Answer #439163 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
Two solutions shown here. I believe there is an infinite number, but I don't know how to prove it yet.
\n" ); document.write( "105,263,157,894,736,842 turns into 052,631,578,947,368,421 when you move the first digit to the end.
\n" ); document.write( "(No one said that the second digit could not be zero).
\n" ); document.write( "210,526,315,789,473,684 turns into 105,263,157,894,736,842
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\n" ); document.write( "HOW I FOUND SOLUTIONS:
\n" ); document.write( "Let \"a\" be the first digit
\n" ); document.write( "The number will be \"a%2A10%5En%2Bb\" where \"b%3C10%5En\" is another natural number
\n" ); document.write( "Moving the first digit to the end yields the number \"10b%2Ba\"
\n" ); document.write( "\"%28a%2A10%5En%2Bb%29%2F2=10b%2Ba\" --> \"a%2A10%5En%2Bb=2%2810b%2Ba%29\" --> \"a%2A10%5En%2Bb=20b%2B2a%29\" --> \"a%2A10%5En-2a=20b-b%29\" --> \"a%2810%5En-2%29=19b\"
\n" ); document.write( "Obviously, \"a\" cannot be a multiple of 19,
\n" ); document.write( "so \"10%5En-2\" must be a multiple of 19.
\n" ); document.write( "\"10%5En-2=19M\" where \"M\" is another natural number
\n" ); document.write( "\"10%5En-2=19M\" --> \"10%5En=19M%2B2\"
\n" ); document.write( "Computer and calculator did not help, so I took paper and pencil.
\n" ); document.write( "I started dividing 1 divided by 19.
\n" ); document.write( "I knew it would be a repeating decimal, and I only needed to get to the 18th digit.
\n" ); document.write( "I was looking for a remainder of 2.
\n" ); document.write( "I got to
\n" ); document.write( "\"1%2F19=0.05263157894736842\" and the remainder digit was 2.
\n" ); document.write( "Multiplying it all by \"10%5E17\" we get
\n" ); document.write( "\"10%5E17%2F19=5263157894736842\" with a remainder of 2,
\n" ); document.write( "which means \"10%5E17=19%2A5263157894736842%2B2\"
\n" ); document.write( "So I found my \"M=5263157894736842\"
\n" ); document.write( "\"10%5En-2=19M\" substituted into \"a%2810%5En-2%29=19b\" means
\n" ); document.write( "\"a%2A19M=19b\" --> \"aM=b\"
\n" ); document.write( "The solutions shown above come from \"a=1\" and \"a=2\".
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