# Math Help - problems

1. ## 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.

2. Hello,
Originally Posted by not happy jan
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 ?

3. Thanks Moo, Have worked out first question but I'm stll having trouble with the 2nd one. Could you please explain a bit more.

4. Originally Posted by not happy jan
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.

5. 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...