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  1. #1
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    problems

    Need help with 2 more questions:
    Find the y-intercept for the linear rule 4x-y=3 and
    What is the last digit in the expansion of 3 to the power of 100. The calculator says the answer to this is 7, but I need to show how I got the answer.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by not happy jan View Post
    Need help with 2 more questions:
    Find the y-intercept for the linear rule 4x-y=3 and
    The y-intercept is the point where the graph intercepts the y-axis.
    Any point on the y-axis has its absissa x=0
    So find the point

    What is the last digit in the expansion of 3 to the power of 100. The calculator says the answer to this is 7, but I need to show how I got the answer.
    Let's say you have a number N, that ends in a number X. N=AB...WX. Then N=10*(AB...W)+X.

    3^1=\textbf{3}
    3^2=\textbf{9}
    3^3=2\textbf{7}=7+2*10
    3^4=\dots \textbf{1}=1+10k
    3^5=\dots \textbf{3}=3+10k'
    3^6=\dots \textbf{9}=9+10k''
    ...

    See the pattern ?
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  3. #3
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    Thanks Moo, Have worked out first question but I'm stll having trouble with the 2nd one. Could you please explain a bit more.
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  4. #4
    Moo
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    Quote Originally Posted by not happy jan View Post
    Thanks Moo, Have worked out first question but I'm stll having trouble with the 2nd one. Could you please explain a bit more.
    What's interesting you is the unit digit, that is to say the last number.
    If you multiply a number whose unit digit is 3 by a number whose unit digit is 7, the unit digit of the result is 1 (*). This explains why I didn't put the exact values of the higher powers of 3.

    And this will help you prove that 3^{4k}, for example, has the same unit digit, whatever the integer k>0 is.

    You can do this by induction or notice that 3^{4k}=(3^4)^k (this one would be easier). But since I don't know your level, it's quite difficult for me to show you a method...

    By the way, the answer is 1, not 7.
    ---------------------------------------------------
    (*)
    Say number N has unit digit n. Then N=n+10*(an integer)
    M has unit digit m. Then M=m+10*(an integer)

    --> M*N=(n+10*(an integer))(m+10*(an integer))=100*(an integer)+10*(an integer)+m*n

    Everything that is multiple of 10 has a unit digit of 0. Thus the unit digit of M*N will be the unit digit of m*n.
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  5. #5
    Kai
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    Yep, the answer is 1, jan, maybe ull understand this..3^100 = 3^(4)*(25)= 81^25,.. and 81 raised to any natural number will always end by 1, test it with calculator if u want, but moo's method is more professional...
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