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Math Help - Problem dealing with some Number Theory "multiple of some #"

  1. #1
    eri
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    Problem dealing with some Number Theory "multiple of some #"

    My Question about this problem:
    They say, 100 is a multiple of 4, so that's why the units digit of 3100 is the same as that of 34 However, 100 is also a multiple of 5. So why did they pick 4?

    Please help explain this--as I am not hot in number theory



    If n is an integer with units digit 3,
    what is the units digit of n 100 ?

    Since only the units digit of n matters, you can start by exploring what happens when n equals 3. Raise 3 to powers and observe the units digits of the results.
    31 = 3
    32 = 9
    33 = 27
    34 = 81
    35 = 243

    units digit:
    3
    9
    7
    1
    3

    Notice that the units digit of 35 is again 3, so from here on the sequence will repeat itself and continue to cycle through the four values 3, 9, 7, 1. Since 100 is a multiple of 4, the units digit of 3100 is the same as that of 34
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by eri View Post
    My Question about this problem:
    They say, 100 is a multiple of 4, so that's why the units digit of 3100 is the same as that of 34 However, 100 is also a multiple of 5. So why did they pick 4?

    Please help explain this--as I am not hot in number theory



    If n is an integer with units digit 3,
    what is the units digit of n 100 ?

    Since only the units digit of n matters, you can start by exploring what happens when n equals 3. Raise 3 to powers and observe the units digits of the results.
    31 = 3
    32 = 9
    33 = 27
    34 = 81
    35 = 243

    units digit:
    3
    9
    7
    1
    3

    Notice that the units digit of 35 is again 3, so from here on the sequence will repeat itself and continue to cycle through the four values 3, 9, 7, 1. Since 100 is a multiple of 4, the units digit of 3100 is the same as that of 34
    at first, i can't understand what you posted.. well, i will edit this for you..

    Quote Originally Posted by eri View Post
    My Question about this problem:
    They say, 100 is a multiple of 4, so that's why the units digit of 3^{100} is the same as that of 3^4 However, 100 is also a multiple of 5. So why did they pick 4?

    Please help explain this--as I am not hot in number theory



    If n is an integer with units digit 3,
    what is the units digit of
    <i>n</i>^{100} ?
    Since only the units digit of n matters, you can start by exploring what happens when n equals 3. Raise 3 to powers and observe the units digits of the results.
    3^1   = 3
    3^2 = 9
    3^3 = 27
    3^4 = 81
    3^5 = 243
    units digit:
    3
    9
    7
    1
    3

    Notice that the units digit of 3^5 is again 3, so from here on the sequence will repeat itself and continue to cycle through the four values 3, 9, 7, 1. Since 100 is a multiple of 4, the units digit of 3^{100} is the same as that of 3^4^
    well, the explanation was given to you..
    note that the unit digit repeats every after 4 powers..
    obeserv also that
    1= 4(0)+1
    2=4(0)+2
    3=4(0)+3
    4=4(1)+0
    5=4(1)+1

    and with powers 1,5,9,... etc, they will have unit digit as 3;
    with powers 2,6,10,... etc, they will have unit digit as 9;
    with powers 3,7,11,... etc, they will have unit digit as 7;
    with powers 4,8,12,... etc, they will have unit digit as 1;

    now, 100=4(25)+0.. and notice that the similarity i has with 4=4(1)+0 is the thing you added to the multiple of 4..

    why pick 4 and why not pick 5? the unit digits repeat every after 4 powers..

    you may compute for 3^{99}.. 99 = 4(24)+ 3.. because of this, the unit digit of 3^{99} is the same as the unit digit of 3^3 which is 7..
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