any help plz

miss blue · #1 $Old$ October 25th, 2009, 12:54 PM

hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .

Amer · #2 $Old$ October 26th, 2009, 12:35 PM

Quote:

Originally Posted by miss blue $View Post$

hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .

let

$a=p_1\times p_2 \times ... \times p_n$

$b=q_1\times q_2 \times ...\times q_m$

such that p,q are prime numbers
g.c.d(a,b) the product of the similar primes in a and b say

$g.c.d(a,b)=p_1\times p_2 \times ... \times p_i=q_1\times q_2 \times ... \times q_i=x^2$

x is integer number, there exist such x since g.c.d is square

$l.c.m(a,b)=g.c.d(a,b)\times (p_{i+1}\times ...\times p_n)(q_{i+1}\times ...\times q_m)=y^2$

y is integer, there exist such y since l.c.m is square

note that
$\left(\frac{a}{g.c.d(a,b)},\frac{b}{g.c.d(a,b)}\right)=1$ in other word

$(p_{i+1}\times ...\times p_n)$ and $(q_{i+1}\times...\times q_m)$ are relatively primes

$g.c.d(a,b)\times (p_{i+1}\times ...\times p_n)(q_{i+1}\times ...\times q_m)=y^2$

g.c.d(a,b) , $(p_{i+1}\times ...\times p_n)$ , $(q_{i+1}\times ...\times q_m)$ all of them are squares

say
$(p_{i+1}\times ...\times p_n)=t^2$

$q_{i+1}\times ...\times q_m)=s^2$

s,t are integers now

$a=x^2(t^2)$

$b=x^2(s^2)$

so a,b are squares

miss blue · #3 $Old$ November 7th, 2009, 06:45 AM

thx alot Amer

but plz if u have a selution for the athors be generous
ant kareem wa n7n nastahal

thx

Amer · #4 $Old$ November 8th, 2009, 08:20 AM

Quote:

Originally Posted by miss blue $View Post$

hi every one
can any buby help me with these questions?

1. If a,b are in N*, if gratest common div.(a,b) & smallest com. mult.(a,b) are squares, show that a,b are squares.

2. Show that for every m>0 , a?1 we have

( a^m -1 / a-1 , a-1 ) = ( a-1 , m ).

3. show that any two terms in the sequance: 2+1, 2^2+1, 2^4+1, .. , 2^2n+1 are coprime .

in the second question you want to prove that

$\left( \frac{a^{m-1}}{a-1} , a-1 \right) = (a-1 , m )$

or

$\left( \frac{a^m -1 }{a-1} , a-1 \right) = ( a-1 , m )$

m>0 , is there any conditions on a

miss blue · #5 $Old$ November 8th, 2009, 08:42 AM

yessss sorry
i copied it wrong

a>1

thx

Amer · #6 $Old$ November 8th, 2009, 09:57 AM

$\left(\frac{a^m-1}{a-1} , a-1 \right) = (a-1 , m )$

note that

$(a,b)=(r,b)$

r is the reminder such that $a=xb+r$

now

$\frac{a^m-1}{a-1} = a^{m-1} + a^{m-2} + ...+ 1$

but

$( a^{m-1} + + ...+ 1 , a-1 ) = (a-1 , r )$

r is the reminder

$a^{m-1} + + ...+ 1 = (a-1)(s) + r$

to find r sub a=1 you will have

$1+1+...+1 = 0(s) + r \Rightarrow r=m$

the proof is finished