# I found new solution for this problem

#### yoyo145

The given is three factors only

#### skipjack

Forum Staff
The proof you provided was for question (7) and it considered successive pairs.

Which question gives only three factors (no "..." to indicate further factors)?

#### yoyo145

I explain no.7 because my friend in school asked me to solve it but in my proof i answered that if we get only three factors like no.8 we able to solve it in convenient way and the student able to understand it

Last edited:

#### yoyo145

finally,i'm able to write on forum

Hi, First of all, I would like to thank you for taking your timeâ€¦â€¦
Prove that: $(1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)(1-Ï‰^8$)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦to 2n factors $=3^n$
In my problems book ,I found this problem. I tried to solve it by myself â€œas usual I canâ€™t solve it at first time â€œ . so I saw my guide answer book . the answer wasnâ€™t clear for me at first time â€œthat is unusual â€ I asked my teacher for that question but he didnâ€™t add anything for me to understand it. so, I search on the internet for explanation but most of them are repeat what my book say !! I donâ€™t lose my hope. I found someone asked the same question in this site https://math.stackexchange.com/questions/2299230/what-is-meaning-of-2n-factors
From the answers on this question I knew The 2n factors mean theâ€ numbers of factors â€œ but I wrote it in my book as a note .
I continue my study normally but my brain canâ€™t stop thinking about that problem
So, I decide to break up my problem â€¦â€¦â€¦â€¦..
I note $(1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)$ they are factors multiply to each other
I began to search â€œhow to multiply factorsâ€â€¦â€¦.after 4 hours
I found that method to multiply factors https://www.youtube.com/watch?v=g4zTQveQuuY
$N^{no.of factors/2}$
Number of factors =2n â€œme: oh yeah!!!â€
What is n ??
we should study factors first
"Factors" are the numbers we can multiply together to get another number:
$2\times{3}$ $=6$
2 and 3 are factors of 6
A number can have many factors.
For example :
$12 =$ $1\times{12}$
$=$ $2\times6$
$=$ $3\times4$
so all factors of â€œ12â€ ={1,2,3,4,6,12} note itâ€™s set.set should have all unique items. We canâ€™t repeat numbers
now I will tell you â€œnâ€=12 â€œthe number that has factorsâ€
if we multiply these factors together we will find =1728
letâ€™s use the formula $N^{no.of factors/2}$
we have 6 factors of 12
$12^{6/2}$ $=1728$
Then, the formula is valid
Before we solve our problem remember these relation to help us simplify our problem
$Ï‰^3$=$1$
$1+Ï‰+$$Ï‰^2$$=0$
$-Ï‰=1+$$Ï‰^2 -Ï‰^2=1+Ï‰ But , wait how to get â€œnâ€ from our problem . we know we get it by multiply two factors But we canâ€™t get 12 by multiply 1 and 2 â€œ1&2 are factors of 12â€ we should find the right two factors to multiply to each otherâ€¦..okay! Letâ€™s solve our problem from this point Prove that: (1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)(1-Ï‰^8)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦to 2n factors =3^n (1-Ï‰) will equal (1+1+$$Ï‰^2)$$=(2+$$Ï‰^2)$
$(1-Ï‰^2)=(1+1+Ï‰)=(2+ Ï‰)$
The next bracket we will find $Ï‰^4=Ï‰^3\times Ï‰=1\times Ï‰=Ï‰$
$(1-Ï‰^4)=(1-Ï‰)=(1+1+Ï‰^2)=(2+Ï‰^2)$
So letâ€™s put them front of us
$(2+Ï‰^2) (2+ Ï‰) (2+Ï‰^2)$â€¦â€¦etc. we will note that two bracket repeatâ€¦..â€two factors repeatâ€
So we have only two factors $(2+Ï‰^2 )(2+Ï‰)$ which they are equal to â€œnâ€â€¦â€¦â€¦â€¦..and number of factors is â€œ2nâ€
Letâ€™s use multiply factors formula $N^{no.of factors/2}$
$((2+Ï‰^2)(2+Ï‰))^{2n/2}$ â€¦â€¦â€¦â€¦2 will go away with 2
$((2+Ï‰^2)(2+Ï‰))^n=(4+2Ï‰+2Ï‰^2+Ï‰^3)^n=(4+2Ï‰+2 Ï‰^2+1)^n=(2+2+2Ï‰+2Ï‰^2+1)^n=(2+2(1+Ï‰+Ï‰^2)+1)^n$
$=3^n$
Thanks $\color{red}{â™¥â™¥}$
Yahia kamal 2019

Last edited: