n belongs to natural no's, then the highest integer m such that 2^m divides 3^2n+2-8n-9 is
=9(9^n)-8n-9
9(8+1)^n -8n - 9
9(nc08n +nc18n-1 + ...........+ncn80) -8n -9
9(8x+1) -8n-9 (let the remaining terms be expressed as x(ie.nc08n-1,nc18n-2 etc.)
72x+9-8n-9
72x-8n
8(9x-n) divisible by 2m
2m =8
m=3
Hallo aditya.
Example: If n = 1 we have:
\(3^{2n+2} - 8n -9 = 3^{2\cdot 1+2} - 8\cdot 1 -9 = 3^4-8-9 = 81-17 = 64 ={\color{red}2^6}\)
\(2^6 \text{ can be devide by } 2^m \text{, if }m=6.\\ So~the~mimimum~ m~ is ~ 6.\\ \text{The maximum m must be equal or greater than the minimum m }.\\ \text{So m is }6\text{ or greater}.\)
m is not a constant.
Hallo aditya.
Example: If n = 1 we have:
\(3^{2n+2} - 8n -9 = 3^{2\cdot 1+2} - 8\cdot 1 -9 = 3^4-8-9 = 81-17 = 64 ={\color{red}2^6}\)
\(2^6 \text{ can be devide by } 2^m \text{, if }m=6.\\ So~the~mimimum~ m~ is ~ 6.\\ \text{The maximum m must be equal or greater than the minimum m }.\\ \text{So m is }6\text{ or greater}.\)
m is not a constant.
Hi Aditya, your answer looks really interesting so I will take a look at your working :)
n belongs to natural no's, then the highest integer m such that 2^m divides 3^2n+2-8n-9 is
=9(9^n)-8n-9
9(8+1)^n -8n - 9
9(nc08n +nc18n-1 + ...........+ncn80) -8n -9
9(8x+1) -8n-9 (let the remaining terms be expressed as x(ie.nc08n-1,nc18n-2 etc.)
72x+9-8n-9
72x-8n
8(9x-n) divisible by 2m
2m =8
m=3
\(3^{2n+2}-8n-9\\ =9*3^{2n}-8n-9\\ =9*9^n-8n-9\\ =9*(8+1)^n-8n-9\\ =9\left[\begin{pmatrix}n\\0\end{pmatrix}8^0+\begin{pmatrix}n\\1\end{pmatrix}8^1+.......\begin{pmatrix}n\\{n-1}\end{pmatrix}8^{n-1}+\begin{pmatrix}n\\n\end{pmatrix}8^n\right]-8^n-9\\ =9\left[1+n*8^1+.......\begin{pmatrix}n\\{n-1}\end{pmatrix}8^{n-1}+8^n\right]-8^n-9\\ =9\left[8n+.......\begin{pmatrix}n\\{n-1}\end{pmatrix}8^{n-1}+8^n\right]-8^n\\\)
I think that is correct but I do not think it simplifies to what you have there.......
In response to aditya below.
You want me to factor 8 out, ok I can do that.
\(=9*8\left[n+\begin{pmatrix}n\\2\end{pmatrix} 8 +........ \begin{pmatrix}n\\n-1\end{pmatrix} 8^{n-2}+8^{n-1} \right]-8 ^n\\ =2^3*9\left[n+\begin{pmatrix}n\\2\end{pmatrix} 8 +........ \begin{pmatrix}n\\n-1\end{pmatrix} 8^{n-2}+8^{n-1} \right]-2^{3n}\\ =2^3*9\left[n+\begin{pmatrix}n\\2\end{pmatrix} 8 +........ \begin{pmatrix}n\\n-1\end{pmatrix} 8^{n-2}+8^{n-1} \right]-2^3*2^{3n-3}\\ =2^3*\left[9\left[n+\begin{pmatrix}n\\2\end{pmatrix} 8 +........ \begin{pmatrix}n\\n-1\end{pmatrix} 8^{n-2}+8^{n-1} \right]-2^{3n-3}\right]\)
So if n>1 and a natural number then this expression is divisable by 8.
Note: if n=1 I don't think that this expression IS divisable by 8. because 8^(-1) is a fraction.
Heureka's simple substitution showed this to be incorrect so there must be something wrong
answer to heureka's comment=
8 also divides 64
you may argue that 4, 2, 1 also divides 64
but i have already proved it above
answer to melody's comment=
when we take 8 common from (nc08^n +nc18^n-1 + ...........+ncn8^0) we get (nc0^8n-1,nc18^n-2....)
for eg- 8 common from (1(nc08^n) we get= 8(8^n-1)
nc18^n-1=8(nc18^n-2) etc
remaining same
what do you think
please reply back
exams are approaching
and binomial theorem carries 90 marks
At Aditya's request, I have continued my analysis on my post #3. BUT it is not correct.
There is an error somewhere. ://
can you please explain where i went wrong in my recent explanation
i had a look at heureka's comment and replied to it
please tell
in the answer book given by IIT(indian institute of technology) which is without explanation, the answer is given as 3
so i thought i got ir right
please elaborate where i went wrong