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.
Hello,
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
Let's say you have a number N, that ends in a number X. N=AB...WX. Then N=10*(AB...W)+X.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.
$\displaystyle 3^1=\textbf{3}$
$\displaystyle 3^2=\textbf{9}$
$\displaystyle 3^3=2\textbf{7}=7+2*10$
$\displaystyle 3^4=\dots \textbf{1}=1+10k$
$\displaystyle 3^5=\dots \textbf{3}=3+10k'$
$\displaystyle 3^6=\dots \textbf{9}=9+10k''$
...
See the pattern ?
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 $\displaystyle 3^{4k}$, for example, has the same unit digit, whatever the integer k>0 is.
You can do this by induction or notice that $\displaystyle 3^{4k}=(3^4)^k$ (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.
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(*)
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.