$3^3 ≡-8 (mod 35)$
$3^3 ≡-2^3 (mod 35)$
$3^6 ≡2^6 (mod 35)$
$3^{6n} ≡2^{6n} (mod 35)$
$3^{6n}-2^{6n} ≡0(mod 35)$
[Proved]
Search found 138 matches
- Fri Sep 14, 2012 8:46 am
- Forum: Number Theory
- Topic: $3^{6n}-2^{6n}$
- Replies: 2
- Views: 2080
- Thu Sep 13, 2012 9:55 pm
- Forum: Algebra
- Topic: SAMO 2012 Problem 1
- Replies: 1
- Views: 1942
Re: SAMO 2012 Problem 1
The summation of $1+3+5+.....$ to n terms is $n^2$
The summation of $2+4+6+.....$ to n terms is $n^2+n$
Now we get-
$\frac {n^2}{n^2+n}=\frac {2011}{2012}$
$2012n^2=2011n^2+2011n$
$n^2=2011n$
$n=2011$
The summation of $2+4+6+.....$ to n terms is $n^2+n$
Now we get-
$\frac {n^2}{n^2+n}=\frac {2011}{2012}$
$2012n^2=2011n^2+2011n$
$n^2=2011n$
$n=2011$
- Thu Sep 13, 2012 3:57 pm
- Forum: Secondary Level
- Topic: when base-power changes......(2)
- Replies: 6
- Views: 4202
Re: when base-power changes......(2)
Problem $(1)$ $3^x-x^3=1$ $3^x = (x+1)(x^2-x+1)$ Here $x≡-1 (mod 3)$. We can write $x=3k-1$. Now we get- $3^{3k-1} = (3k-1+1){(3k-1)^2-3k+1+1}$ $3^{3k-1} = 3k(9k^2-9k+3)$ $3^{3k-3} = k(3k^2-3k+1)$ Here only prime factor of $k(3k^2-3k+1)$ is $3$. So either $k$ or $(3k^2-3k+1)$ is $1$. When, $(3k^2-3k...
- Wed Sep 12, 2012 11:26 pm
- Forum: Geometry
- Topic: SAMO 2012 Problem 2
- Replies: 4
- Views: 3108
Re: SAMO 2012 Problem 2
Structure
- Wed Sep 12, 2012 11:22 pm
- Forum: Geometry
- Topic: SAMO 2012 Problem 2
- Replies: 4
- Views: 3108
SAMO 2012 Problem 2
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$ . The lines $AX$ and $BX$ intersect $CD$ in $X$ and $Y$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac {2} {3}$ , Determine the length of $YZ$.
- Tue Sep 11, 2012 10:40 pm
- Forum: Secondary Level
- Topic: Number Theory Problems
- Replies: 4
- Views: 3453
Number Theory Problems
Problems from the book - “ যারা গণিত অলিম্পিয়াডে যাবে ” 1. If $n,m$ are integers, prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5≠33$. 2. If $n$ is an integer, prove that $n^2+3n+5$ is not divisible by $121$. 3. If $n$ is an integer, prove that $(20^n ≠ 16^n-3^n-1)$ is divisible by $323$. 4. If ...
- Tue Sep 11, 2012 12:46 pm
- Forum: Combinatorics
- Topic: Probability :Two gamblers
- Replies: 3
- Views: 3586
Probability :Two gamblers
Two gamblers A and B play a game throwing two ordinary dice. A wins if he obtains the sum 6 before B obtains 7, and B wins if he obtains the sum 7 before A obtains 6. Which of the players has a better chance of winning if player A starts the game? Show your full solution so that it can be understood...
- Tue Sep 11, 2012 12:29 pm
- Forum: Geometry
- Topic: সহজ সমাধান চাই
- Replies: 2
- Views: 2717
Re: সহজ সমাধান চাই
There is a solution which I got from my friend. It's big. :arrow: We can see , $\angle ACB = 180-(10+70)-(60+20) = 20°$ $\angle AEB = 180-70-(60+20) = 30°$ Draw a line from point D parallel to AB, labeling the intersection with BC as a new point F and conclude: $▲DCF ≈▲ACB$ $\angle CFD = \angle CBA ...
- Sun Sep 09, 2012 4:44 pm
- Forum: International Olympiad in Informatics (IOI)
- Topic: গণিত সমাচার
- Replies: 4
- Views: 5548
- Fri Sep 07, 2012 3:22 pm
- Forum: International Olympiad in Informatics (IOI)
- Topic: IOI-2012 Bangladesh team
- Replies: 7
- Views: 6397
Re: IOI-2012 Bangladesh team
Best of luck. May success be your friend there.