# Number Theory/Expansions

1. May 8, 2007

### f(x)

1. The problem statement, all variables and given/known data
a)True or False :
$$\underbrace{111111111.....1}$$ is a prime number .
$$91 times$$

b) Find n such that -:
$$2\times 2^2+ 3\times2^3+4\times2^4+ \cdots + n\times2^n=2^{n+10}$$

2. Relevant equations

3. The attempt at a solution

I have no idea about a). The number is not divisible by 3,7,11...but i cant go on dividing all the way like this. How do i resolve this into prime factors(if possible) ? Do i use binomial theorem and how ?

About b), LHS is an AGP . I tried taking the $$\ n\times2^n$$ to RHS and then dividing by 2^2...but that doesnt seem to help ?

2. May 10, 2007

### f(x)

any suggestions ?

3. May 10, 2007

### NateTG

Here are some hints...
1.
a) 91 is composite
b) Perhaps you can start by taking partial sums and seeing what happens...

4. May 14, 2007

### f(x)

Umm i dont understand...how does 91 being composite influence divisibility ?
Could you plz explain in abit more detail
Thx

5. May 14, 2007

### mjsd

one reason that it helps is because 91=7x13 means that you can chop the number into segments like
$$\underbrace{\underbrace{1111111}_{7\; times}\underbrace{1111111}_{7\; times}\ldots\underbrace{1111111}_{7\; times}}_{13\; times}$$

and if 1111111 is not a prime then you are done. Otherwise more work needed.

6. May 14, 2007

### Gib Z

$$11 + 1100 + 11 0000 + 11 000000 .......$$

O god damn it, 91's an odd number nvm me didn't notice

Last edited: May 14, 2007
7. May 14, 2007

### NateTG

111,111,111,111,111=111*1,001,001,001,001

8. May 23, 2007

### truewt

For problem (b) you might want to consider the sum to n terms of
$$1+2x+3x^2+4x^3+...+nx^{n-1}$$

9. May 23, 2007

### truewt

If i am not mistaken (b)'s answer is $$2^9+1$$
but you need to do the workings yourself..

10. May 25, 2007

### f(x)

How did you get that ?

Yeah 513 was surely an option...

11. May 25, 2007

### Gib Z

1111111 = 239 x 4649