document.write( "Question 477751: what is the unit digit of 7raisedto 155 how to solve it? \n" ); document.write( "

Algebra.Com's Answer #852396 by ikleyn(52946) $\"\"$ $\"About$

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\n" ); document.write( "What is the unit digit of 7 raised to 155 ? How to solve it?
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\n" ); document.write( "\n" ); document.write( "        The problems of this type are very popular. \r
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\n" ); document.write( "\n" ); document.write( "        You often can find them in tests, in Math competitions, in Math Olympiads, in entrance exams \r
\n" ); document.write( "\n" ); document.write( "        and even in the interview.\r
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\n" ); document.write( "\n" ); document.write( "        Therefore, to be successful in all these aspects, you need to know VERY FIRMLY\r
\n" ); document.write( "\n" ); document.write( "        HOW TO solve such problems.\r
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\n" ); document.write( "\n" ); document.write( "        If such a problem is given to you, the right solution should bounce off you \r
\n" ); document.write( "\n" ); document.write( "        like a ball bounces off a wall.\r
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\n" ); document.write( "\n" ); document.write( "- - - - - - - - S O L U T I O N - - - - - - - - \r
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document.write( "If you will rise your number 7 in consecutive degrees, you will find very soon, that the last \r\n" );
document.write( "digit is repeating.\r\n" );
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document.write( "This is not accidental - it is natural.\r\n" );
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document.write( "Indeed, every time you rise to the next degree, the last digit is a result of multiplication \r\n" );
document.write( "of the previous last digit by 7.\r\n" );
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document.write( "But the number of possible last digits is limited: it is only 10, because there are 10 digits,\r\n" );
document.write( "from 0 to 9, in all.\r\n" );
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document.write( "Therefore inevitably, some last digit will repeat one of preceding last digits.\r\n" );
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document.write( "As soon as it will happen for the first time, the digits will be repeating cyclically.\r\n" );
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document.write( "Let's consider how it works in our case. I will not write whole many-digit numbers,\r\n" );
document.write( "since there is no need in it. I only will write and track the last digit.\r\n" );
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document.write( "    degree k          1    2    3    4    5    6    7    8    9\r\n" );
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document.write( "    the last digit    7    9    3    1    7    9    3    1    7\r\n" );
document.write( "    of  \r\n" );
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document.write( "You see that first time repeating happens at k = 5,  and after k=4 the cycle (7,9,3,1)  is repeating again and again.\r\n" );
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document.write( "So, for  ,  there are many repeating cycles of the length 4 - they are not interesting for us.\r\n" );
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document.write( "The number 155 gives the remainder  3  when divided by 4:  155 = 4*38 + 3.\r\n" );
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document.write( "It tells us that the sough last digit in the number    is the third term in the basic  cycle of four digits  {7, 9, 3, 1}.\r\n" );
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document.write( "From it, we conclude that the last digit of the number    is 3, since '3' is the third term of the cycle {7, 9, 3, 1}.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is HOW TO such problems should be solved and analyzed.\r
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