Problems

[tushar7]

Since no has not posted any problems ...............i am giving some

$1.$ what is the unit digit of $13^{2010}$ ?
$2.$ $m$ points are taken in each sides of a ragular $n$ gon .what is the total number of straight lines that can be drawn using all those lines ?

[Zzzz]

What means unit digit?

[HandaramTheGreat]

What means unit digit?

একক স্থানীয় অংক

[rakeen]

1. It should be 9!
2. I know thatthis prob has to be solved by combinatorix. But ami ekhono combinatorix sikhi nai, so....

[tushar7]

1. It should be 9!
2. I know thatthis prob has to be solved by combinatorix. But ami ekhono combinatorix sikhi nai, so....

its better to post the logic behind your solution

[Zzzz]

Probably Rakeen will give solution of problem 1.

Here is solution to problem 2:
If we choose any two points that are not on same side of the $n\ gon$ then we will find a straight line.
Lets number the sides of $n\ gon$ with $1,2,3,...,n$.
Start with side $1$. For each of the $m$ points of this side, we have $(n-1)m$ points. So in total $(n-1)m^2$ lines.
For each point of side #$2$, we have $(n-2)m$ points. So $(n-2)m^2$ lines.
Thus, total number of line is\[(n-1)m^2+(n-2)m^2+(n-3)m^2+...+1\cdot m^2\] \[=\frac{n(n-1)}{2}\cdot m^2\]

[AntiviruShahriar]

1.ans is 9
logic 13^{fi(10)}_=1(mod10)
13^4 _=1(mod10)
13^2008 _=1(mod10)
13^2008+2_=13^2_=9(mod10)
ঠিক আছে?
2.solution টাও বুঝি নাই।কারণ আমি combinatorics অতো ভাল বুঝি না।
এখানে _= হচ্ছে is congruent to

[Moon]

শাহরিয়ার, ফোরামে স্বাগতম।
আশা করি লক্ষ্য করেছ যে এই ফোরামে সমীকরণ লেখার চমৎকার সুবিধ আছে। তুমিও ইচ্ছা করলে কিভাবে LaTeX দিয়ে লিখতে হয় এই টপিকটা পড়ে সমীকরণ লিখতে পার।

[rakeen]

Ami congurance ekhono sikhi nai. Ei jonno ami ebhabe ber korsi:

$3^1$ er ekok sthanio onko 3
$3^2$ er ekok sthanio onko 9
$3^3$ er ekok sthanio onko 7
$3^4$ er ekok sthanio onko 1

and 2010 is divided by 2, not by 4. So the answer is 9!

[AntiviruShahriar]

ভাই exclamatory sign use কইরো না প্লিজ।এইটায় ৩৬২৮৮০ বুঝায়।@রাকীন।